# 4. FIRST STEPS IN THE THEORY 4.1. A

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 4. FIRST STEPS IN THE THEORY 4.1. A Catalogue of All Groups: The Impossible Dream The fundamental problem of group theory is to systematically explore the landscape and to chart what lies out there. We d like to have a catalogue of all groups, or perhaps just all finite groups: a catalogue that would provide one specimen of each group so that we d be able to say that every group, or every finite group, is isomorphic to exactly one of the specimens in our catalogue. In particular we d know exactly how many groups there are of any given order. This is an impossible dream, but not simply because there are infinitely many finite groups. After all there are infinitely many finite-dimensional vector spaces over a given field yet we can describe them all in the single statement that every finite-dimensional vector space over a field F is isomorphic to the space of n-tuples (x 1, x 2,..., x n ) over F for some n. This is just a corollary of the theorem that every finite-dimensional vector space has a basis. In other words there s exactly one vector space over F (up to isomorphism) for each dimension. Structure theorems exist in various parts of mathematics. We know all the finite fields. In topology the compact surfaces are all known. Within group theory itself we have a number of classification theorems. For example we know all the finite abelian groups, via the Fundamental Theorem of Abelian Groups that we ll prove in a later chapter. The most celebrated example is the classification of all finite simple groups. Don t worry now what a simple group is. In a certain sense they re the building blocks of all finite groups but the term simple is deceptive. They re not the most elementary of groups but their simplicity refers to the fact that they can t be pulled apart, in a certain way, into smaller groups. They re the atoms of the universe of finite groups. Now this celebrated theorem is the culmination of virtually a century s work by many thousands of group theorists and its proof is scattered over numerous research journals in the literature. It has never been assembled in one place because it s estimated that it would occupy about 20,000 pages and because of this it has even made it to the Guinness Book of Records as the theorem with the longest proof! The goal of this chapter is to develop enough of the theory to enable us to prove a couple of classification theorems. Nothing as ambitious as the one we were just talking about just a couple of nice, gentle theorems. Together they describe, for each prime p, the groups of order p and of order 2p. There aren t very many such groups for a given prime. In fact there s only one group of order p and just two of order 2p. So contrary to what you might expect the number of different groups doesn t grow steadily with the group order. The groups of order 32 have been catalogued and there are over 50 of them. But there s only one group of order 31 because 31 is prime and there are just two groups of order Additive and Multiplicative Notation Until now we've written the combination of a and b in an abstract group as a * b. But, unless there's danger of confusion, it's more usual to write the binary operation as if it was addition or multiplication. If we re proving theorems about abstract groups in general, where the group could be non-abelian, we use multiplicative notation. If the groups we re considering are all abelian we normally use additive notation. The reason for this distinction is that we re used to 75

2 multiplication being non-commutative (for example matrices and permutations), but in all the systems we've ever encountered, addition is commutative. It s very dangerous therefore to use additive notation in a non-commutative group. GENERAL MULTIPLICATIVE ADDITIVE NOTATION NOTATION NOTATION a * b ab a + b a * a *... * a a n na e 1 0 inverse of a a 1 a Unless otherwise stated, we ll use multiplicative notation Basic Properties of Groups We begin by showing that each group has only one identity element. Once proven this will allow us to use the notation 1 for the identity (or 0 in abelian groups). Theorem 1: The identity element of a group is unique. Proof: Suppose e, f are identities for a group G. Then e = e f (since f is an identity) = f (since e is an identity). Before we can start using the notation a 1 (or a in abelian groups) we must ensure that every element has a unique inverse. Theorem 2: Each element of a group has only one inverse. Proof: Suppose b, c are both inverses of the element a in a group G. Then b = b1 = b (ac) = (ba) c = 1c = c. Theorem 3: (Cancellation Law): If ax = ay then x = y. Proof: Suppose ax = ay. Then a 1 (ax) = a 1 (ay). Hence (a 1 a)x = (a 1 a)y. Thus 1x = 1y and so x = y. Notice that all the group axioms (except the Closure Law) are needed to prove this. Similarly one can prove that xa = ya implies that x = y. Thus cancellation on the left, or on the right, is possible. A consequence of the cancellation law is that: For a finite group every element appears exactly once and only once in each row and column of the multiplication table. You should take careful note of the fact that you can t cancel on the left-hand end of one side of the equation and on the right-hand end of the other. In other words ax = xb does not imply that a = b Powers If g is an element of a group, we define positive integer powers of g inductively as follows: g 0 = 1; g n+1 = g n g for all n 0. We define negative powers by g n = (g 1 ) n for all negative integers n. 76

3 Theorem 4: For all natural numbers m, n and all elements a, b in a group G: (1) a m a n = a m+n ; (2) (a m ) n = a mn ; (3) if G is abelian, (ab) n = a n b n ; (4) (b 1 ab) n = b 1 a n b. Proof: Although these seem obvious enough (and indeed they are obvious if m and n are positive, just by counting factors) the cases where one or both are negative require special attention. (1) This is obvious if both m, n are positive or zero. Suppose m is positive and n is negative, say n = r. If m r then on the left hand side there will be r cancelling pairs of aa 1 leaving m r = m + n factors. If m < r there will be only m such pairs leaving r m factors of a 1. The result is therefore a (rm) = a mr = a m+n. We ve thus proved the result for all n where m 0. If m is negative, say m = s, then putting b = a 1 the left hand side is b s b n. By the earlier case this is b sn = a ns = a m+n. (2) Again this is obvious if m, n are both positive. The other cases are left as an exercise. (3) If n is positive we simply count the number of factors on each side. Because the group is assumed to be abelian the factors may be rearranged so all the a's can be brought to the front. If n is zero, LHS = RHS = 1. If n is negative we put b = a 1 and use the positive case. (4) If n is positive, (b 1 ab) n = b 1 (bb 1 ) a(bb 1 )... (bb 1 ) ab (n factors) = b 1 aa... ab = b 1 a n b. If n = 0, LHS = RHS = 1. If n is negative we use the positive case on b 1 cb where c is defined to be a 1. Theorem 5: In a group G (ab) 1 = b 1 a 1. Proof: (b 1 a 1 )(ab) = b 1 (a 1 a) b = 1 and similarly (ab)(b 1 a 1 ) = 1. Remember that the inverse of a product is the product of the inverses in reverse order. The cyclic subgroup generated by an element g is the set of all powers of g (including 1 as g 0 and negative powers). It s denoted by g. Definition: The order of an element g of a group is the smallest positive integer n such that g n = 1 (if such an n exists). In additive notation this becomes ng = 0. The order of g is denoted by g. If there s no such positive n, we say that g has infinite order. For example the order of i in the group of non-zero complex numbers under multiplication is 4 since i 4 = 1 while no lower positive power is equal to 1. The number 2 has infinite order. The identity element of any group is the only element of order 1. Theorem 6: The order of an element g G is at most G. Proof: Let n = G. The elements 1, g, g 2,... g n1, g n can t be distinct (there are n+1 of them in a group of order n). Hence there must be some repetition: g r = g s for some r, s with 0 r < s n. Thus g sr = 1 and since 0 < r < s n, g n. Later, we ll prove that, in fact, the order of an element of G divides the order of G. Theorem 7: Groups of even order must contain an element of order 2. Proof: Suppose G is even. Now the elements of G that differ from their inverses must come in pairs {x, x 1 }. Since G is even, the remaining elements, those for which x = x 1, 77

4 must also be even in number. But x = x 1 is equivalent to x 2 = 1 and so these are the elements of order 2, together with the identity. Leaving out the identity, there must be an odd number of elements of order 2 and so the number of elements of order 2 must be at least 1. Theorem 8: If all of the elements of G (except 1) have order 2, then G must be abelian. Proof: Let x, y G. Then (xy) 2 = 1. But also x 2 y 2 = 1 and so xyxy = xxyy. Multiplying by x 1 on the left and by y 1 on the right of each side of the equation we conclude that yx = xy. Since this holds for all x, y G it follows that G is abelian Cyclic Groups A group G is cyclic if it can be generated by a single element, that is if G = g for some g G. Example 1: The set of n-th roots of unity, G = {z C z n = 1}, is a group under multiplication. It is a cyclic group because it can be generated by e 2i/n. This is because every n-th root of 1 has the form e 2ki/n = (e 2i/n ) k. In particular the group of 4-th roots of unity is {1, i, 1, i} which is generated by i. Example 2: The group of symmetries of a parallelogram is {I, R} where R is a 180 rotation about its centre. This group is generated by R. Example 3: The group (Z, +) of integers under addition is cyclic because it can be generated by the integer 1. Remember that we re using additive notation here so instead of saying that every integer is an integer power of 1 (which is not the case), we should be saying that every integer is an integer multiple of 1 (which it is). Note that 1 also generates this group, but ±1 are the only generators. Theorem 9: Cyclic groups are abelian. Proof: Two typical elements in the cyclic group g are g r and g s. Now g r g s = g r+s = g s g r. So every pair of elements commute and hence the cyclic group is abelian. Theorem 10: A finite group of order n is cyclic if and only if it contains an element of order n. Proof: If G = n and G has an element g of order n then g, the cyclic subgroup generated by g has order n (that is there are n distinct powers of g). Thus there are no other elements in G. They re all powers of g and so g is a generator for G and hence G is cyclic. Conversely suppose that G is a cyclic group of order n. Then, if g is a generator, g must have order n. 78

5 Example 4: The group given by the following multiplication table isn t cyclic since it has no element of order 4. A B C D A A B C D B B A D C C C D A B D D C B A 4.6. Subgroups If G is a group and H is a subset of G then H is a subgroup if (1) xy H for all x, y H (2) 1 H (3) x 1 H for all x H. A subgroup H is a group in its own right. Every group is a subgroup of itself. All other subgroups are called proper subgroups. The trivial subgroup, 1 = {1} is a subgroup of any group. Notation: H G means H is a subgroup of G and H < G means H is a proper subgroup of G, that is, H is a subgroup but is not G itself. Example 5: 2Z (the group of even integers) is a proper subgroup of Z (under +). Theorem 11: For all g G, g is a subgroup of G. Proof: For all r,s Z +, g r g s = g r+s g; 1 = g 0 g and for all r, (g r ) 1 = g r g Cosets and Lagrange s Theorem Let H G. Define a relation by defining x y if x = yh for some h H. Theorem 12: is an equivalence relation. Proof: Reflexive Let a G. Then a = a1. Since 1 H, a a. Symmetric Suppose a b. Then a = bh for some h H. Hence ah 1 = b. Since h 1 H, b a. Transitive Suppose a b and b c. Then a = bh for some h H and b = ck for some k H. Thus a = (ck)h = c(kh). Since kh H, a c. NOTE: Each of the three properties of an equivalence relation comes from one of the three closure properties of a subgroup. The equivalence classes under are called the right cosets of H in G. Notation: gh denotes the right coset containing g. NOTE: Left cosets are defined similarly. Note also that some books define left and right cosets in the opposite manner. Example 6: G = R #, H = {±1} under the operation of multiplication. The cosets are all of the form {±x}. Example 7: G = C #, the group of non-zero complex numbers under multiplication and let H be the set of all complex numbers with modulus 1. The right (or left) cosets of H in G are all the circles with centre O. 79

6 Example 8: Suppose G = S 3 and H = {I, (12)}. The left and right cosets of H in G are: HI = {I.I, (12)I} = {I, (12)} IH = {I.I, I(12)} = {I, (12)} H(13) = {I(13), (12)(13)} = {(13), (123)} (13)H = {(13)I, (13)(12)} = {(13), (132)} H(23) = {I(23), (12)(23)} = {(23), (132)} (23)H = {(23)I, (23)(12)} = {(23), (123)}. Notice that in this case, left and right cosets are different. Example 9: G = S 3, H = {I, (123), (132)}. Here the left and right cosets give the same decomposition of G. These two (left/right) cosets are H itself and {(12), (13), (23)}. Theorem 13: (1) The subgroup H is itself one of the cosets of H in G. (2) Two elements a, b belong to the same right coset (of H in G) if and only if b 1 a H. Proof: (1) H = 1H. (2) a, b belong to the same right coset if and only if a b, that is, if and only if a = bh for some h H, that is, if and only if b 1 a = h for some h H, or more simply, b 1 a H. Equivalence classes in general can be of different sizes, but cosets of a given subgroup must have the same size. Theorem 14: If H G, every coset of H in G has H elements. Proof: There s a natural 1-1 correspondence between any coset ah and H itself viz. ah h. Hence the number of elements in each is the same. Theorem 15: (Lagrange) The order of a subgroup (of a finite group) divides the order of the group. Proof: Suppose there are m cosets of H in G. Since G is the disjoint union of them and each coset has H elements, it follows that G = m. H and so H divides G. This is a very powerful result. It shows that the number of elements in a group greatly affects its structure. Example 10: If G = 14, the only possible orders for a subgroup are 1, 2, 7 and 14. Theorem 16: Groups of prime order are cyclic. Proof: Suppose G = p where p is prime. Since p 2 we may choose g G such that g 1. Let H = g. Let H = n. Now n p and n > 1 so n = p. Hence G = H and so is cyclic. Since the order of an element is the order of the cyclic subgroup it generates, we have: Theorem 17: The order of an element of a finite group divides the order of the group. Lagrange s Theorem is a powerful one. But its converse does not hold in general. Just because a number divides the group order does not mean there has to be a subgroup of that order. For example A 4 has order 12, but no subgroup of order 6. However if a prime power divides G there is at least one subgroup of that order. This is part of what is known as the Sylow Theorems. Here we prove the special case where the divisor is just a prime. 80

7 Theorem 18 (Cauchy): If p is a prime divisor of G there exists an element of G of order p. Proof: Consider all equations of the form g 1 g 2 g p = 1 where the g i G. There are G p1 such equations, because the first p 1 factors can be chosen arbitrarily and the last one has to be the inverse of their product. Now if g 1 g 2 g p = 1 is one of these equations then so is g 2 g 3 g p g p1 = 1. In fact we can bring any number of the factors from the left-hand side and bring them to the right. Any cyclic rearrangement of the factors will also give one of these equations. You might think that the set of these equations can be decomposed into sets of size p in this way. But what if every factor is equal? For example the cyclic rearrangements of the equation = 1 will give just one equation not p distinct equations. While ever two of the factors are different the cyclic rearrangements will be distinct. Notice that it is important for p to be prime for this to work because if p = 6 then ababab = 1 would only have two distinct rearrangement. Now if G has no element of order p then = 1 is the only equation that is by itself. All the others can be decomposed into sets of size p. But this would mean that the number of equations of the above form is congruent to 1 modulo p, which is impossible for G p1. [Remember that G > 1, so this is not p 0 and so it is genuinely a multiple of p.] 4.8. Dihedral Groups The family of cyclic groups contains those groups with the simplest possible group structure. A closely related family is the family of dihedral groups. The dihedral group of order 2n is the group: D 2n = A,BA n = 1, B 2 = 1, BA = A 1 B. Dihedral groups occur naturally in many different guises. D 2n is, for example, the group structure of the symmetry group of a regular n-sided polygon. The symmetry operations consist of the rotation R through 2/n, and its powers plus the 180 rotations about the n axes of symmetry. But if Q denotes any one of these, the others can be expressed in the form R k QR k. If Q denotes the 180 degree rotation about any one axis of symmetry (say a vertical axis), any other 180 symmetry operation can be achieved by rotating that axis until it becomes vertical (by some power R k ), carrying out Q about the vertical axis, and then rotating the axis back to its original position (by R k ). Hence it can be expressed as R k QR k. Now R n = 1 (n successive rotations through 2/n); Q 2 = 1 (two successive 180 rotations) and QRQ = R 1 (a negative or clockwise rotation can be achieved by rotating about the vertical axis first, then rotating in the positive direction and finally rotating back in the vertical axis try it!) and so RQ = QR 1. So this symmetry group is R, Q R n = 1, Q 2 = 1, RQ = QR 1, that is, it is the dihedral group of order 2n. In particular D 8 is the symmetry group of a square. 81

8 4.9. Dihedral Arithmetic The dihedral group D 2n = A,BA n = 1, B 2 = 1, BA = A 1 B has three relations. Let s examine their implications. A n = 1: This means that any expression involving A's and B's need not have any string of successive A's longer than n1, because any block of n successive A's is A n which, because it s equal to the identity, can be removed. For example in D 4 = A,BA 4 = 1, B 2 = 1, BA = A 1 B, an element such as AABAAAAAAABA can be simplified to AABAAABA = A 2 BA 3 BA by removing an A 4 from the middle. B 2 = 1: This means that it is never necessary to have two successive B's. For example in D 8 an expression such as ABBBAAABBAA can be simplified to ABAAAAA by using B 2 = 1, twice. This can be further reduced to ABA by use of the relation A 4 = 1. Another consequence of B 2 = 1 is B = B 1 (just multiply both sides on the left by B 1 ). This means that there s never any need to have B 1 in any expression. BA = A 1 B: It is this third relation that makes dihedral groups non-abelian (except for the trivial cases of D 4 and D 2 where A 1 = A). Expressing this relation in words, we can say that every time a B passes across an A it inverts it, that is, converts it to A 1. Consequently if we have an expression involving a mixture of A's and B's we can move all the A's up to the front and all the B's down to the back just as we would if the commutative law was in force. The difference is that the A's get inverted every time a B crosses over. This is the dihedral twist. Example 13: The expression ABA 3 BA 2 BAB can be written as: AA 3 BBA 2 BAB = A 2 BBA 2 BAB = A 2 A 2 BAB = BAB = A 1 B 2 = A 1. Theorem 21: The elements of D 2n are: 1 A A 2 A 3 A n1 B AB A 2 B A 3 B A n1 B Proof: Because of the relation BA = A 1 B we can express every element in the form A i B j. But because A n = 1 and B 2 = 1, we may assume that i = 0, 1, 2,..., n 1 and j = 0 or 1. Notice that the first row consists of the cyclic subgroup, H, generated by A, and the second row is the left coset HB. Theorem 22: In the dihedral group A,BA n = 1, B 2 = 1, BA = A 1 B the elements of the form A k B all have order 2. Proof: (A k B) 2 = A k BA k B = A k A k BB = BB = 1. 82

9 4.10. Groups of Order 2p We now classify groups of order 2p (where p is prime). We show that all such groups are either cyclic or dihedral. Since the proof is rather lengthy we ve broken the argument up into 39 steps. Also, to make it easier to see which assumptions are in force at any given time, we ve used a similar indenting convention to that advocated when writing computer programs. Theorem 23: If G = 2p for some prime p then G is cyclic or dihedral. Proof: (1) Suppose that G = 2p where p is prime (2) Suppose that p is odd. (3) By Lagrange's theorem the order of every element is 1, 2, p or 2p. (4) Suppose G contains an element of order 2p. (4) Hence G is cyclic. (5) Since G is even, G must contain an element, b, of order 2. (6) Suppose that all the elements of G, except 1, have order 2. (7) Then G is abelian. (8) Choose a, b G of order 2 with a b. (9) Hence H = {1, a, b, ab} is a subgroup of G of order 4. (10) Since 4 doesn t divide 2p we get a contradiction. (11) So G must have an element, a, of order p. (12) Let H = a. (13) Since 2 doesn t divide p, b H. (14) The right cosets of H in G must be H and bh. (15) Similarly the left cosets are H and Hb. (16) Since bh and Hb consist of all the elements outside H, they must be equal, ie. Hb = bh. (17) Now ba bh so ba Hb. (18) Hence ba = a r b for some integer r = 0, 1,..., p 1. (19) Hence b 2 a = ba r b (20) = baa... ab (where there are r factors of a). (21) = a r a r... a r b 2 (where there are r factors of a r ). (22) = a r² (since b 2 = 1). (23) So a r² = a, (24) Hence p divides r 2 1. (25) Since p is prime and r 2 1 = (r 1)(r + 1), pr1 or pr+1. (26) Thus ba = ab or a 1 b. (27) Suppose ba = ab. (28) Now the order of ab must be 1, 2 or p. (29) Suppose ab = 1. (30) Then a = b 1 = b, a contradiction. (31) Suppose ab has order 2. (32) Then 1 = (ab) 2 = a 2 b 2 = a 2, a contradiction. (33) Suppose ab has order p. (34) Then 1 = (ab) p = a p b p = b p, a contradiction. (35) So b 1 ab = a 1 and so G = a,ba p = b 2 = 1, ba = a 1 b = D 2p. (36) Suppose that p = 2. (37) Then every element of G (except 1) has order 2. 83

10 (38) Hence G is abelian. (39) Hence G = a,ba 2 = b 2 = 1, ba = ab = a,ba 2 = b 2 = 1, ba = a 1 b = D 4. EXERCISES FOR CHAPTER 4 1 a b EXERCISE 1: Show that the set of all matrices over Z 2 of the form 0 1 c is a group of order 8 under matrix multiplication. Does it satisfy the commutative law? EXERCISE 2: Find all the elements of order 4 in the above group. EXERCISE 3: Show that the above group is a dihedral group of order 8, that is find an element A of order 4 and an element of order 2 such that BA = A 1 B. EXERCISE 4: Find all the elements of order 4 in Z 2 Z 8. EXERCISE 5: How many elements of the group C # of non-zero complex numbers under multiplication have order 7? How many have order 8? EXERCISE 6: The following is a partially completed group table for a group. Complete it. a b c d a a b c c a d d Which element is the identity? EXERCISE 7: Find all 4 possible group tables for a group G = {1, a, b, c} of order 4. Show that three of them are isomorphic (meaning that any one of them can be transformed to any other by a suitable relabelling) and the fourth is fundamentally different (eg look at the number of elements of order 2). EXERCISE 8: If G is the group whose table is given below, show that H = {1, c, d} and K = {1, b} are both subgroups of G. Find all the left and right cosets of each subgroup. 1 a b c d e 1 1 a b c d e a a 1 c b e d b b d 1 e a c c c e a d 1 b d d b e 1 c a e e c d a b 1 EXERCISE 9: If G = a,b a 4 = 1, b 3 = 1, ab = ba and H is the cyclic subgroup generated by b, find the right and left cosets of H in G. 84

11 EXERCISE 10: In the dihedral group D 10 = a, b a 5 = 1, b 2 = 1, ab = ba 1, simplify a 7 b 3 a 2 baba 3 ba 2 a 7 b 2 a. EXERCISE 11: Simplify a 13 b 5 a 2 b 7 a 2 ba in the above group D 10. EXERCISE 12: Find the order of H given the following clues: (a) H is a proper subgroup of a group of order 68 (b) H is non-cyclic EXERCISE 13: Find the order of H given the following clues. (a) H is a subgroup of some group of order 100. (b) H contains no element of order 2. (c) H is not cyclic. EXERCISE 14: Find the order of H given the following clues: (a) H is a subgroup of some group G of order 168. (b) H is a subgroup of another group K of order 112. (c) H is not cyclic or dihedral. (d) H contains an element of order 7. (e) H has more than two left cosets in K. EXERCISE 15: If G is a group, the centre (zentrum in German) of G is defined to be Z(G) = {gg gx = xg for all xg}. In other words, it s the set of all elements that commute with everything. (a) Prove that Z(G) is a subgroup of G. (b) Find Z(D 2n ). [HINT: You ll need to consider odd and even values of n separately.] EXERCISE 16: Let G = {x Rx 1} denote the set of real numbers excluding 1, and let the binary operation * be defined on G by a * b = a + b + ab (i) Prove that (G,*) is a group. (ii) What is the inverse of 2? (iii) How many elements does G have of order 2? EXERCISE 17: Find the numbers of elements of each order in the following two groups whose group tables are given: (a) 1 a b c 1 1 a b c a a 1 c b b b c 1 a c c b a 1 (b) 1 a b c 1 1 a b c a a b c 1 b b c 1 a c c 1 a b 85

12 EXERCISE 18: Find the orders of the elements in the cyclic group of order 6. EXERCISE 19: Find the orders of the elements of the following three groups: G = the group {0,1,2,3,4,5,6,7} under addition modulo 8; H = the group {1, 3, 7, 9, 11, 13, 17, 19} under multiplication modulo 20; K = the dihedral group of order 8. Show that no two of these groups are isomorphic. EXERCISE 20: Which of the above three groups of order 8 is cyclic? Which are abelian? EXERCISE 21: Find 9, the cyclic group generated by 9, in the group Z 100 #. This consists of all the integers from 1 to 99 that have no factors in common with 100. The operation is multiplication modulo 100. Also determine the order of 9 in this group. EXERCISE 22: Find the order of the following elements in the group Z 100 (consisting of all the integers from 0 to 99 under addition modulo 100): 2, 9, 6, 15. EXERCISE 23: Find the left and right cosets of {1, b} in the dihedral group D 12 = a,b a 6 = 1, b 2 = 1, ab = ba 1. EXERCISE 24: G is the group whose table is given below. Show that H = {1, a, d, f} and K = {1, d} are both subgroups of G. Find all the left and right cosets of each subgroup. 1 a b c d e f g 1 1 a b c d e f g a a d e g f c 1 b b b g d 1 c a e f c c e 1 d b f g a d d f c b 1 g a e e e b f a g d c 1 f f 1 g e a b d c g g c a f e 1 b d EXERCISE 25: Find the order of H given the following clues: (a) H is a proper subgroup of a group of order 52; (b) H is non-abelian. EXERCISE 26: Find the order of H given the following clues: (a) H is a subgroup of some group G of order 100. (b) H is a subgroup of another group K of order 40. (c) H is not cyclic or dihedral. EXERCISE 27: Find the order of H given the following clues: (a) H is a subgroup of some group G of order 20. (b) H is non-abelian. (c) G contains an element g of order 2 and an element h of order 5. (d) H contains h but not g. 86

13 EXERCISE 28: If G is a group, the derived subgroup of G is defined to be the subgroup generated by all the elements of the form x 1 y 1 xy as x, y range over the elements of the group. Find the derived subgroup of D 2n. [HINT: You will need to consider odd and even values of n separately.] EXERCISE 29: Complete the following group table and find the orders of the elements. 1 a b c d e 1 a 1 c b d b d a 1 c d e a b d c e b a e b a c 1 EXERCISE 30: Find the numbers of elements of each order in the cyclic group of order 12. EXERCISE 31: Find the order of the following elements in the group Z # 17 the integers from 1 to 16 under multiplication modulo 17): 2, 6, 9, 16 (consisting of all EXERCISE 32: Find the orders of the elements of the following three groups: G = the group {1, i, (1 + i)/2, (1 i)/2} under multiplication; H = the group of symmetries of a rectangular box; K = the group of order 8 whose group table is: 1 1 i i j j k k i i j j k k i i j j k k i i i 1 1 k k j j i i i 1 1 k k j j j j j k k 1 1 i i j j j k k 1 1 i i k k k j j i i 1 1 k k k j j i i 1 1 Show that no two of these groups are isomorphic. SOLUTIONS FOR CHAPTER 4 EXERCISE 1: 1 a b 1 a b 1 a + a b + b + ac There are 2 3 = 8 matrices of this form, over Z c c = 0 1 c + c so the set is closed under matrix multiplication. The associative law holds, as it always does 1 a b with matrix multiplication. The identity matrix has this form. Finally the inverse of 0 1 c

14 1 a ac b is 0 1 c which has the required form. Finally = while = so the group doesn t satisfy the commutative law EXERCISE 2: , EXERCISE 3: Take A = and B = EXERCISE 4: The elements of Z 2 Z 8 are vectors of the form (x, y) where x Z 2 and y Z 8. If 4(x, y) = (0, 0) then 4x = 0 and 4y = 0. This places no restriction on x: x = 0 or 1 but 4y = 0 means y = 0, 2, 4 or 6. If 2(x, y) = (0, 0) then 2y = 0. So we must exclude y = 0 and y = 4. The elements of order 4 are thus (0, 2), (0, 6), (1, 2) and (1, 6). EXERCISE 5: There are 6 elements of order 7, namely e i/7,, e 6i/7 and 4 elements of order 8, namely e i/8, e 3i/8, e 5i/8, e 7/8. EXERCISE 6: Since bc = c the element b is the identity. a b c d a a b a b c d c c a d d Now dc cannot be any of a, c or d because that would result in a repetition in either the row or the column corresponding to d. So dc must be b. In a similar way we can complete the table: a b c d a b a d c b a b c d c d c a b d c d b a 88

15 EXERCISE 7: Filling out the entries for the identity we get: 1 a b c 1 1 a b c a a b b c c Now ab = 1 or c. Case I: ab = 1: We can now complete the group table: 1 a b c 1 1 a b c a a c 1 b b b 1 c a c c b a 1 Case II: ab = c: The group table is thus: 1 a b c 1 1 a b c a a c b b c c Now bc = 1 or a. Case IIA: bc = 1: The group table is thus: 1 a b c 1 1 a b c a a c b b c a 1 c c 1 If a 2 = b then ac = 1, a contradiction. Hence a 2 = 1 and so the group table is: 1 a b c 1 1 a b c a a 1 c b b b c a 1 c c b 1 a Case IIB: bc = a: The group table is: 1 a b c 1 1 a b c a a c b b c 1 a c c a 89

16 This can be completed in two possible ways: 1 a b c 1 a b c 1 1 a b c 1 1 a b c a a 1 c b a a b c 1 b b c 1 a b b c 1 a c c b a 1 c c 1 a b Of these four possibilities the ones that arise in Cases I and IIA are isomorphic to the second possibility under Case IIB. They therefore represent a single group with differing notation. This group has just one element of order 2. The remaining possibility (the first under Case IIB) is essentially different in that it has 3 elements of order 2. So there are two groups of order 4. Note that both of them are abelian. EXERCISE 8: The fact that H and K are subgroups of G can be most easily seen by extracting their group tables from the main table: H 1 c d 1 1 c d c c d 1 d d 1 c K 1 b 1 1 b b b 1 Since every entry in each table belongs to the subset in each case each subset is closed under multiplication. Clearly each contains the identity and, since 1 appears in each row and column, every element has an inverse within the respective subset. The left cosets of H in G are: H = {1, c, d} and ah = (a1, ac, ad} = {a, b, e}. The right cosets of H in G are: H = {1, c, d} and Ha = {1a, ca, da} = {a, e, b}. NOTE that in this example the left and right cosets are the same, even though the group is non-abelian. The left cosets of K in G are: K = {1, b}, ak = {a1, ab} = {a, c} and dk = {d1, db} = {d, e}. NOTE that we didn't waste our time with bk or ck because those elements were already included and we would have simply repeated the first two cosets. For example bk = {b1, bb} = {b, 1} = {1, b}. So always use as a representative for a new coset, an element that hasn t yet been included. 90

17 The right cosets of K in G are: K = {1, b}, Ka = {1a, ba} = {a, d} and Kc = {1c, bc} = {c, e} NOTE that in this case the left cosets and the right cosets give two different subdivisions of the group. EXERCISE 9: The elements are: 1, a, a 2, a 3, b, ab, a 2 b, a 3 b, b 2, ab 2, a 2 b 2, a 3 b 2. The left cosets are H = {1, b, b 2 } ah = {a, ab, ab 2 } a 2 H = {a 2, a 2 b, a 2 b 2 } and a 3 H = {a 3, a 3 b, a 3 b 2 } Of course since this group is abelian the left cosets are the same as the right cosets. EXERCISE 10: Using b 2 = 1 this becomes a 7 ba 2 baba 3 ba 2 a 7 a, and combining powers of a we get a 7 ba 2 baba 3 ba 10. Using a 5 = 1 we get a 2 ba 3 baba 3 b. Now we need to make use of the relation ab = ba 1. Moving a b past an a, inverts it. Moving the second last b to the back we get a 2 ba 3 baa 3 b 2 = a 2 ba 3 ba 3. Moving the next b down gives a 2 ba 3 a 3 b = a 2 bb = a 2. EXERCISE 11: The expression can be successively simplified as follows: a 13 b 5 a 2 b 7 a 2 ba = a 3 ba 2 ba 2 ba = a 3 ba 2 ba 2 a 1 b = a 3 ba 2 bab = a 3 ba 2 a 1 = a 3 ba = a 2 b. EXERCISE 12: By Lagrange's Theorem H divides 68. Since H isn t cyclic and groups of prime order are cyclic, H isn t prime. Similarly it isn t 1 (groups of order 1 contain just the identity and are cyclic). So we re looking for a divisor of 68 that is neither 1 nor prime. Since the prime factorisation of 68 is , the only possibilities are 4 and 68. However since H is a proper subgroup, this means it s not the whole of G and so 68 is ruled out. Thus, by a process of elimination, H = 4. EXERCISE 13: By Lagrange's Theorem H divides 100. H being non-cyclic rules out 1 and the primes 2 and 5, leaving 4, 5, 10, 20, 25, 50 and 100. Now groups of even order must contain an element of order 2. Since H doesn t, it must have odd order, leaving 25 as the only possibility. EXERCISE 14: By Lagrange s Theorem, H divides both 168 = 837 and 112 = 167 and therefore must divide their greatest common divisor, which is 56. Since H contains an element of order 7, H must be divisible by 7. This limits the possibilities to 7, 14, 28 and 56. Now since H is neither cyclic nor dihedral, H can t be prime or twice a prime [groups of order p are cyclic; groups of order 2p are cyclic or dihedral]. This narrows down the possibilities to 28 and 56. Now if H was 56 there would be exactly 2 left cosets in K which has order 112. By clue (e) this isn t so, and hence 56 is ruled out. Therefore H must be

18 EXERCISE 15: (a) This is easily verified. Note that gx = xg implies that xg 1 = g 1 x. (b) D 2n = a,ba n =b 2 =1, ba= a 1 b. If a r Z(G) then a r b = ba r = a r b, so a 2r = 1 which means that n2r. If n is odd this means nr and so a r = 1. If n is even a r is 1 or a n/2. Similarly one can check that no element of the form a r b commutes with a. So Z(D 2n ) = {1} if n is odd and {1, a n/2 } if n is even. EXERCISE 16: Closure: Let a, b G, so a 1 and b 1. Suppose a * b = 1. Then a + b + ab = 1 and so (a + 1)(b + 1) = 0 which implies that a = 1 or b = 1, a contradiction. Associativity: Unlike the examples in exercise 1, this is a totally new operation that we have never encountered before. We must therefore carefully check the associative law. (a * b) * c = (a * b) + c + (a * b) c = ab + a + b + c + (ab + a + b)c = a + b + c + ab + ac + bc + abc Similarly a * (b * c) has the same value (we can actually see this by the symmetry of the expression. Identity: An identity, e, would have to satisfy: e * x = x = x * e for all x G, that is, ex + e + x = x, or e(x + 1) = 0 for all x. Clearly e = 0 works. The identity is 0. Inverses: If x * y = 0, then xy + x + y = 0. So y(x + 1) = x and hence y = x/(x + 1). This exists for all x 1, i.e. for all x G. But we must also check that it is itself an element of G. Suppose x/(x + 1) = 1. Then x = x 1, which gives 0 = 1. This is clearly a contradiction and hence y must be in G. EXERCISE 17: (a) has the 1 plus 3 elements of order 2. (b) has 1, one element of order 2 (viz. b) and 2 elements of order 4. [The fact that they differ in their structure in this way means that they re non-isomorphic, or essentially different. These two tables reflect the only two possible group structures for a group of order 4.] EXERCISE 18: The cyclic group of order 6 has the form: {1, g, g 2, g 3, g 4, g 5 } where g 6 = 1 Clearly g has order 6. (g 2 ) 2 = g 4, (g 2 ) 3 = g 6 = 1 and so g 2 has order 3. (g 3 ) 2 = 1 and so g 3 has order 2. (g 4 ) 2 = g 8 = g 2, (g 4 ) 3 = g 12 = 1 and so g 4 has order 3; The powers of g 5 are g 5, g 10 = g 4, g 15 = g 3, g 20 = g 2, g 25 = g. Finally (g 5 ) 6 = g 30 = 1 and so g 5 has order 6. The cyclic group of order 6 thus has: 1 element of order 1; 1 element of order 2; 2 elements of order 3; 2 elements of order 6. NOTE: orders 4 and 5 are missed out. Can you guess why? 92

19 EXERCISE 19: G: 1, 3, 5, 7 have order 8 2, 6 have order 4 4 has order 2 0 has order 1 H: 3, 7, 13 and 17 have order 4 9, 11 and 19 have order 2 1 has order 1 K: The dihedral group a,ba 4 = b 2 = 1, ab = ba 1 has elements: 1, a, a 2, a 3, b, ab, a 2 b, a 3 b. Of these: a, a 3 have order 4 a 2, b, ab, a 2 b, a 3 b have order 2; 1 has order 1. NOTE: The order of the group is a power of 2 and the orders of the elements are likewise powers of 2. This is quite significant. Listing the numbers of the elements of orders 1, 2, 4 and 8 as vectors we have: G: (1,1,2,4); H: (1,3,4,0); K: (1,5,2,0) The differences show that these three groups are mutually non-isomorphic. There are in fact 5 distinct groups of order 8, the above three plus two others. EXERCISE 20: G is cyclic (and hence abelian) because it contains an element of order 8; H is abelian but not cyclic; K is non-abelian (and hence not cyclic). NOTE: There s always only one cyclic group of any given order. In other words, all cyclic groups of order n are isomorphic. A representative example of the cyclic group of order n is Z n = {0, 1,... n1} of integers modulo n under addition. EXERCISE 21: The powers of 9 are: 9 1 = 9, 9 2 = 81, 9 3 = 729 = 29 mod 100 and so on. Rather than accumulate the higher and higher powers we can simply multiply by 9 at each stage to get the next: 9 4 = 9 29 = 261 = 61 Then comes 49, 41, 69, 21, 89 and finally 1. So 9 = {1, 9, 81, 29, 61, 49, 41, 69, 21, 89}. There are 10 elements in this cyclic subgroup and so 9 has order 10 under multiplication modulo 100. EXERCISE 22: 2: Remember that the operation is addition, so we need to keep adding the generator to itself, that is, taking higher and higher multiples, not powers. We want the smallest positive integer n such that 2n = 0 mod 100, or in other words, such that 2n is a multiple of 100. The answer is clearly 50. 9: We want 100 to divide 9n. Since 100 has no factors in common with 9, we d need n itself to be a multiple of 100. The smallest positive such n is thus 100. So 9 has order 100 in this group. 6: We need 6n to be a multiple of 100. Since 2 divides both 6 and 100 we need 50 to divide 3n. But since 50 has no factor in common with 3, we'd need 50 to divide n. So 6 has order

20 15: 15n = 0 mod 100 means 3n = 0 mod 20. Since 3 is coprime with 20, we need n = 0 mod 20, so 15 has order 20. EXERCISE 23: Left cosets: Right cosets i.e. 1 b a ab a 2 a 2 b a 3 a 3 b a 4 a 4 b a 5 a 5 b 1 b a ba a 2 ba 2 a 3 ba 3 a 4 ba 4 a 5 ba 5 1 b a a 5 b a 2 a 4 b a 3 a 3 b a 4 a 2 b a 5 ab EXERCISE 24: H and K contain 1 and are closed under multiplication and inverse as these tables show: H 1 a d f 1 1 a d f a a d f 1 d d f 1 a f f 1 a d K 1 d 1 1 a d d 1 The left and right cosets of H are 1 a d f b c e g The left and right cosets of K are 1 d a f b c e g Both H and K are normal subgroups. EXERCISE 25: By (a) H = 1, 2, 4, 13, 26 and by (b) H = 26. EXERCISE 26: By (a) H divides 100. By (b) H divides 40. Hence H divides = 20. Hence H = 1, 2, 4, 10 or 20. By (c) H = 20. EXERCISE 27: By (a) H divides 20. By (c), (d) H is divisible by 5. Hence H = 5, 10 or 20. By (b) H 5. By (d) H 20. Hence H = 10. EXERCISE 28: Let G = D 2n = a, b a n = b 2 = 1, ba = a 1 b. Since a 1 b 1 ab = a 2, a 2 G and hence G contains all even powers of a. Case I: n is even. Let H = a 2. Since b 1 a 2k b = a 2k H, H is a normal subgroup of G. Since G/H = 4, G/H is abelian and so G H. Thus G = H = a 2. 94

### GROUPS ACTING ON A SET

GROUPS ACTING ON A SET MATH 435 SPRING 2012 NOTES FROM FEBRUARY 27TH, 2012 1. Left group actions Definition 1.1. Suppose that G is a group and S is a set. A left (group) action of G on S is a rule for

### I. GROUPS: BASIC DEFINITIONS AND EXAMPLES

I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called

### Solutions to TOPICS IN ALGEBRA I.N. HERSTEIN. Part II: Group Theory

Solutions to TOPICS IN ALGEBRA I.N. HERSTEIN Part II: Group Theory No rights reserved. Any part of this work can be reproduced or transmitted in any form or by any means. Version: 1.1 Release: Jan 2013

### = 2 + 1 2 2 = 3 4, Now assume that P (k) is true for some fixed k 2. This means that

Instructions. Answer each of the questions on your own paper, and be sure to show your work so that partial credit can be adequately assessed. Credit will not be given for answers (even correct ones) without

### Basic Properties of Rings

Basic Properties of Rings A ring is an algebraic structure with an addition operation and a multiplication operation. These operations are required to satisfy many (but not all!) familiar properties. Some

### Elements of Abstract Group Theory

Chapter 2 Elements of Abstract Group Theory Mathematics is a game played according to certain simple rules with meaningless marks on paper. David Hilbert The importance of symmetry in physics, and for

### ORDERS OF ELEMENTS IN A GROUP

ORDERS OF ELEMENTS IN A GROUP KEITH CONRAD 1. Introduction Let G be a group and g G. We say g has finite order if g n = e for some positive integer n. For example, 1 and i have finite order in C, since

### Elementary Number Theory We begin with a bit of elementary number theory, which is concerned

CONSTRUCTION OF THE FINITE FIELDS Z p S. R. DOTY Elementary Number Theory We begin with a bit of elementary number theory, which is concerned solely with questions about the set of integers Z = {0, ±1,

### Chapter 7: Products and quotients

Chapter 7: Products and quotients Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 42, Spring 24 M. Macauley (Clemson) Chapter 7: Products

3. QUADRATIC CONGRUENCES 3.1. Quadratics Over a Finite Field We re all familiar with the quadratic equation in the context of real or complex numbers. The formula for the solutions to ax + bx + c = 0 (where

### 13 Solutions for Section 6

13 Solutions for Section 6 Exercise 6.2 Draw up the group table for S 3. List, giving each as a product of disjoint cycles, all the permutations in S 4. Determine the order of each element of S 4. Solution

### SUBGROUPS OF CYCLIC GROUPS. 1. Introduction In a group G, we denote the (cyclic) group of powers of some g G by

SUBGROUPS OF CYCLIC GROUPS KEITH CONRAD 1. Introduction In a group G, we denote the (cyclic) group of powers of some g G by g = {g k : k Z}. If G = g, then G itself is cyclic, with g as a generator. Examples

### 10. Graph Matrices Incidence Matrix

10 Graph Matrices Since a graph is completely determined by specifying either its adjacency structure or its incidence structure, these specifications provide far more efficient ways of representing a

### 1 Symmetries of regular polyhedra

1230, notes 5 1 Symmetries of regular polyhedra Symmetry groups Recall: Group axioms: Suppose that (G, ) is a group and a, b, c are elements of G. Then (i) a b G (ii) (a b) c = a (b c) (iii) There is an

### 2. Let H and K be subgroups of a group G. Show that H K G if and only if H K or K H.

Math 307 Abstract Algebra Sample final examination questions with solutions 1. Suppose that H is a proper subgroup of Z under addition and H contains 18, 30 and 40, Determine H. Solution. Since gcd(18,

### Further linear algebra. Chapter I. Integers.

Further linear algebra. Chapter I. Integers. Andrei Yafaev Number theory is the theory of Z = {0, ±1, ±2,...}. 1 Euclid s algorithm, Bézout s identity and the greatest common divisor. We say that a Z divides

### INTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS

INTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS STEVEN HEILMAN Contents 1. Homework 1 1 2. Homework 2 6 3. Homework 3 10 4. Homework 4 16 5. Homework 5 19 6. Homework 6 21 7. Homework 7 25 8. Homework 8 28

### Department of Mathematics Exercises G.1: Solutions

Department of Mathematics MT161 Exercises G.1: Solutions 1. We show that a (b c) = (a b) c for all binary strings a, b, c B. So let a = a 1 a 2... a n, b = b 1 b 2... b n and c = c 1 c 2... c n, where

### December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS

December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in two-dimensional space (1) 2x y = 3 describes a line in two-dimensional space The coefficients of x and y in the equation

### 5.1 Commutative rings; Integral Domains

5.1 J.A.Beachy 1 5.1 Commutative rings; Integral Domains from A Study Guide for Beginner s by J.A.Beachy, a supplement to Abstract Algebra by Beachy / Blair 23. Let R be a commutative ring. Prove the following

### MODULAR ARITHMETIC. a smallest member. It is equivalent to the Principle of Mathematical Induction.

MODULAR ARITHMETIC 1 Working With Integers The usual arithmetic operations of addition, subtraction and multiplication can be performed on integers, and the result is always another integer Division, on

### 6 Commutators and the derived series. [x,y] = xyx 1 y 1.

6 Commutators and the derived series Definition. Let G be a group, and let x,y G. The commutator of x and y is [x,y] = xyx 1 y 1. Note that [x,y] = e if and only if xy = yx (since x 1 y 1 = (yx) 1 ). Proposition

### some algebra prelim solutions

some algebra prelim solutions David Morawski August 19, 2012 Problem (Spring 2008, #5). Show that f(x) = x p x + a is irreducible over F p whenever a F p is not zero. Proof. First, note that f(x) has no

### Solutions to Assignment 4

Solutions to Assignment 4 Math 412, Winter 2003 3.1.18 Define a new addition and multiplication on Z y a a + 1 and a a + a, where the operations on the right-hand side off the equal signs are ordinary

### GROUP ALGEBRAS. ANDREI YAFAEV

GROUP ALGEBRAS. ANDREI YAFAEV We will associate a certain algebra to a finite group and prove that it is semisimple. Then we will apply Wedderburn s theory to its study. Definition 0.1. Let G be a finite

### Notes on Algebraic Structures. Peter J. Cameron

Notes on Algebraic Structures Peter J. Cameron ii Preface These are the notes of the second-year course Algebraic Structures I at Queen Mary, University of London, as I taught it in the second semester

### Abstract Algebra Cheat Sheet

Abstract Algebra Cheat Sheet 16 December 2002 By Brendan Kidwell, based on Dr. Ward Heilman s notes for his Abstract Algebra class. Notes: Where applicable, page numbers are listed in parentheses at the

### CLASSIFYING FINITE SUBGROUPS OF SO 3

CLASSIFYING FINITE SUBGROUPS OF SO 3 HANNAH MARK Abstract. The goal of this paper is to prove that all finite subgroups of SO 3 are isomorphic to either a cyclic group, a dihedral group, or the rotational

### Applications of Fermat s Little Theorem and Congruences

Applications of Fermat s Little Theorem and Congruences Definition: Let m be a positive integer. Then integers a and b are congruent modulo m, denoted by a b mod m, if m (a b). Example: 3 1 mod 2, 6 4

### Math 313 Lecture #10 2.2: The Inverse of a Matrix

Math 1 Lecture #10 2.2: The Inverse of a Matrix Matrix algebra provides tools for creating many useful formulas just like real number algebra does. For example, a real number a is invertible if there is

GENERATING SETS KEITH CONRAD 1 Introduction In R n, every vector can be written as a unique linear combination of the standard basis e 1,, e n A notion weaker than a basis is a spanning set: a set of vectors

### Group Theory (MA343): Lecture Notes Semester I Dr Rachel Quinlan School of Mathematics, Statistics and Applied Mathematics, NUI Galway

Group Theory (MA343): Lecture Notes Semester I 2013-2014 Dr Rachel Quinlan School of Mathematics, Statistics and Applied Mathematics, NUI Galway November 21, 2013 Contents 1 What is a group? 2 1.1 Examples...........................................

### Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm.

Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm. We begin by defining the ring of polynomials with coefficients in a ring R. After some preliminary results, we specialize

### MATH 433 Applied Algebra Lecture 13: Examples of groups.

MATH 433 Applied Algebra Lecture 13: Examples of groups. Abstract groups Definition. A group is a set G, together with a binary operation, that satisfies the following axioms: (G1: closure) for all elements

### G = G 0 > G 1 > > G k = {e}

Proposition 49. 1. A group G is nilpotent if and only if G appears as an element of its upper central series. 2. If G is nilpotent, then the upper central series and the lower central series have the same

### Revised Version of Chapter 23. We learned long ago how to solve linear congruences. ax c (mod m)

Chapter 23 Squares Modulo p Revised Version of Chapter 23 We learned long ago how to solve linear congruences ax c (mod m) (see Chapter 8). It s now time to take the plunge and move on to quadratic equations.

### CHAPTER 5: MODULAR ARITHMETIC

CHAPTER 5: MODULAR ARITHMETIC LECTURE NOTES FOR MATH 378 (CSUSM, SPRING 2009). WAYNE AITKEN 1. Introduction In this chapter we will consider congruence modulo m, and explore the associated arithmetic called

### On the generation of elliptic curves with 16 rational torsion points by Pythagorean triples

On the generation of elliptic curves with 16 rational torsion points by Pythagorean triples Brian Hilley Boston College MT695 Honors Seminar March 3, 2006 1 Introduction 1.1 Mazur s Theorem Let C be a

### Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

### Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

### Group Theory. Contents

Group Theory Contents Chapter 1: Review... 2 Chapter 2: Permutation Groups and Group Actions... 3 Orbits and Transitivity... 6 Specific Actions The Right regular and coset actions... 8 The Conjugation

### Quotient Rings and Field Extensions

Chapter 5 Quotient Rings and Field Extensions In this chapter we describe a method for producing field extension of a given field. If F is a field, then a field extension is a field K that contains F.

### Chapter 7. Permutation Groups

Chapter 7 Permutation Groups () We started the study of groups by considering planar isometries In the previous chapter, we learnt that finite groups of planar isometries can only be cyclic or dihedral

### Mathematics Course 111: Algebra I Part II: Groups

Mathematics Course 111: Algebra I Part II: Groups D. R. Wilkins Academic Year 1996-7 6 Groups A binary operation on a set G associates to elements x and y of G a third element x y of G. For example, addition

### Inverses and powers: Rules of Matrix Arithmetic

Contents 1 Inverses and powers: Rules of Matrix Arithmetic 1.1 What about division of matrices? 1.2 Properties of the Inverse of a Matrix 1.2.1 Theorem (Uniqueness of Inverse) 1.2.2 Inverse Test 1.2.3

### A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions

A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions Marcel B. Finan Arkansas Tech University c All Rights Reserved First Draft February 8, 2006 1 Contents 25

### Binary Operations. Discussion. Power Set Operation. Binary Operation Properties. Whole Number Subsets. Let be a binary operation defined on the set A.

Binary Operations Let S be any given set. A binary operation on S is a correspondence that associates with each ordered pair (a, b) of elements of S a uniquely determined element a b = c where c S Discussion

### Congruences. Robert Friedman

Congruences Robert Friedman Definition of congruence mod n Congruences are a very handy way to work with the information of divisibility and remainders, and their use permeates number theory. Definition

### Notes on finite group theory. Peter J. Cameron

Notes on finite group theory Peter J. Cameron October 2013 2 Preface Group theory is a central part of modern mathematics. Its origins lie in geometry (where groups describe in a very detailed way the

### COMMUTATIVE RINGS. Definition: A domain is a commutative ring R that satisfies the cancellation law for multiplication:

COMMUTATIVE RINGS Definition: A commutative ring R is a set with two operations, addition and multiplication, such that: (i) R is an abelian group under addition; (ii) ab = ba for all a, b R (commutative

### k, then n = p2α 1 1 pα k

Powers of Integers An integer n is a perfect square if n = m for some integer m. Taking into account the prime factorization, if m = p α 1 1 pα k k, then n = pα 1 1 p α k k. That is, n is a perfect square

### Math 312 Homework 1 Solutions

Math 31 Homework 1 Solutions Last modified: July 15, 01 This homework is due on Thursday, July 1th, 01 at 1:10pm Please turn it in during class, or in my mailbox in the main math office (next to 4W1) Please

### UNIT 2 MATRICES - I 2.0 INTRODUCTION. Structure

UNIT 2 MATRICES - I Matrices - I Structure 2.0 Introduction 2.1 Objectives 2.2 Matrices 2.3 Operation on Matrices 2.4 Invertible Matrices 2.5 Systems of Linear Equations 2.6 Answers to Check Your Progress

### The Dirichlet Unit Theorem

Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if

### Groups, Rings, and Fields. I. Sets Let S be a set. The Cartesian product S S is the set of ordered pairs of elements of S, S S = {(x, y) x, y S}.

Groups, Rings, and Fields I. Sets Let S be a set. The Cartesian product S S is the set of ordered pairs of elements of S, A binary operation φ is a function, S S = {(x, y) x, y S}. φ : S S S. A binary

### Mathematics of Cryptography

CHAPTER 2 Mathematics of Cryptography Part I: Modular Arithmetic, Congruence, and Matrices Objectives This chapter is intended to prepare the reader for the next few chapters in cryptography. The chapter

### Definition A square matrix M is invertible (or nonsingular) if there exists a matrix M 1 such that

0. Inverse Matrix Definition A square matrix M is invertible (or nonsingular) if there exists a matrix M such that M M = I = M M. Inverse of a 2 2 Matrix Let M and N be the matrices: a b d b M =, N = c

### Algebraic Systems, Fall 2013, September 1, 2013 Edition. Todd Cochrane

Algebraic Systems, Fall 2013, September 1, 2013 Edition Todd Cochrane Contents Notation 5 Chapter 0. Axioms for the set of Integers Z. 7 Chapter 1. Algebraic Properties of the Integers 9 1.1. Background

### ABSTRACT ALGEBRA: A STUDY GUIDE FOR BEGINNERS

ABSTRACT ALGEBRA: A STUDY GUIDE FOR BEGINNERS John A. Beachy Northern Illinois University 2014 ii J.A.Beachy This is a supplement to Abstract Algebra, Third Edition by John A. Beachy and William D. Blair

### The determinant of a skew-symmetric matrix is a square. This can be seen in small cases by direct calculation: 0 a. 12 a. a 13 a 24 a 14 a 23 a 14

4 Symplectic groups In this and the next two sections, we begin the study of the groups preserving reflexive sesquilinear forms or quadratic forms. We begin with the symplectic groups, associated with

### 1 VECTOR SPACES AND SUBSPACES

1 VECTOR SPACES AND SUBSPACES What is a vector? Many are familiar with the concept of a vector as: Something which has magnitude and direction. an ordered pair or triple. a description for quantities such

### r + s = i + j (q + t)n; 2 rs = ij (qj + ti)n + qtn.

Chapter 7 Introduction to finite fields This chapter provides an introduction to several kinds of abstract algebraic structures, particularly groups, fields, and polynomials. Our primary interest is in

### Problem Set 1 Solutions Math 109

Problem Set 1 Solutions Math 109 Exercise 1.6 Show that a regular tetrahedron has a total of twenty-four symmetries if reflections and products of reflections are allowed. Identify a symmetry which is

### Notes on Factoring. MA 206 Kurt Bryan

The General Approach Notes on Factoring MA 26 Kurt Bryan Suppose I hand you n, a 2 digit integer and tell you that n is composite, with smallest prime factor around 5 digits. Finding a nontrivial factor

### Chapter 13: Basic ring theory

Chapter 3: Basic ring theory Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 42, Spring 24 M. Macauley (Clemson) Chapter 3: Basic ring

### COSETS AND LAGRANGE S THEOREM

COSETS AND LAGRANGE S THEOREM KEITH CONRAD 1. Introduction Pick an integer m 0. For a Z, the congruence class a mod m is the set of integers a + mk as k runs over Z. We can write this set as a + mz. This

### Linear Codes. Chapter 3. 3.1 Basics

Chapter 3 Linear Codes In order to define codes that we can encode and decode efficiently, we add more structure to the codespace. We shall be mainly interested in linear codes. A linear code of length

### The noblest pleasure is the joy of understanding. (Leonardo da Vinci)

Chapter 6 Back to Geometry The noblest pleasure is the joy of understanding. (Leonardo da Vinci) At the beginning of these lectures, we studied planar isometries, and symmetries. We then learnt the notion

### Introduction to Finite Fields (cont.)

Chapter 6 Introduction to Finite Fields (cont.) 6.1 Recall Theorem. Z m is a field m is a prime number. Theorem (Subfield Isomorphic to Z p ). Every finite field has the order of a power of a prime number

### GROUP ACTIONS ON SETS WITH APPLICATIONS TO FINITE GROUPS

GROUP ACTIONS ON SETS WITH APPLICATIONS TO FINITE GROUPS NOTES OF LECTURES GIVEN AT THE UNIVERSITY OF MYSORE ON 29 JULY, 01 AUG, 02 AUG, 2012 K. N. RAGHAVAN Abstract. The notion of the action of a group

### Group Fundamentals. Chapter 1. 1.1 Groups and Subgroups. 1.1.1 Definition

Chapter 1 Group Fundamentals 1.1 Groups and Subgroups 1.1.1 Definition A group is a nonempty set G on which there is defined a binary operation (a, b) ab satisfying the following properties. Closure: If

### SPECIAL PRODUCTS AND FACTORS

CHAPTER 442 11 CHAPTER TABLE OF CONTENTS 11-1 Factors and Factoring 11-2 Common Monomial Factors 11-3 The Square of a Monomial 11-4 Multiplying the Sum and the Difference of Two Terms 11-5 Factoring the

### Kevin James. MTHSC 412 Section 2.4 Prime Factors and Greatest Comm

MTHSC 412 Section 2.4 Prime Factors and Greatest Common Divisor Greatest Common Divisor Definition Suppose that a, b Z. Then we say that d Z is a greatest common divisor (gcd) of a and b if the following

### ENUMERATION BY ALGEBRAIC COMBINATORICS. 1. Introduction

ENUMERATION BY ALGEBRAIC COMBINATORICS CAROLYN ATWOOD Abstract. Pólya s theorem can be used to enumerate objects under permutation groups. Using group theory, combinatorics, and many examples, Burnside

### S on n elements. A good way to think about permutations is the following. Consider the A = 1,2,3, 4 whose elements we permute with the P =

Section 6. 1 Section 6. Groups of Permutations: : The Symmetric Group Purpose of Section: To introduce the idea of a permutation and show how the set of all permutations of a set of n elements, equipped

### MATH 304 Linear Algebra Lecture 4: Matrix multiplication. Diagonal matrices. Inverse matrix.

MATH 304 Linear Algebra Lecture 4: Matrix multiplication. Diagonal matrices. Inverse matrix. Matrices Definition. An m-by-n matrix is a rectangular array of numbers that has m rows and n columns: a 11

### Assignment 8: Selected Solutions

Section 4.1 Assignment 8: Selected Solutions 1. and 2. Express each permutation as a product of disjoint cycles, and identify their parity. (1) (1,9,2,3)(1,9,6,5)(1,4,8,7)=(1,4,8,7,2,3)(5,9,6), odd; (2)

### Elementary Number Theory and Methods of Proof. CSE 215, Foundations of Computer Science Stony Brook University http://www.cs.stonybrook.

Elementary Number Theory and Methods of Proof CSE 215, Foundations of Computer Science Stony Brook University http://www.cs.stonybrook.edu/~cse215 1 Number theory Properties: 2 Properties of integers (whole

### a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.

Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given

### 3. Equivalence Relations. Discussion

3. EQUIVALENCE RELATIONS 33 3. Equivalence Relations 3.1. Definition of an Equivalence Relations. Definition 3.1.1. A relation R on a set A is an equivalence relation if and only if R is reflexive, symmetric,

### Cryptography and Network Security. Prof. D. Mukhopadhyay. Department of Computer Science and Engineering. Indian Institute of Technology, Kharagpur

Cryptography and Network Security Prof. D. Mukhopadhyay Department of Computer Science and Engineering Indian Institute of Technology, Kharagpur Module No. # 01 Lecture No. # 12 Block Cipher Standards

### 3 1. Note that all cubes solve it; therefore, there are no more

Math 13 Problem set 5 Artin 11.4.7 Factor the following polynomials into irreducible factors in Q[x]: (a) x 3 3x (b) x 3 3x + (c) x 9 6x 6 + 9x 3 3 Solution: The first two polynomials are cubics, so if

### Introduction to Modern Algebra

Introduction to Modern Algebra David Joyce Clark University Version 0.0.6, 3 Oct 2008 1 1 Copyright (C) 2008. ii I dedicate this book to my friend and colleague Arthur Chou. Arthur encouraged me to write

### U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009. Notes on Algebra

U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009 Notes on Algebra These notes contain as little theory as possible, and most results are stated without proof. Any introductory

### 4. CLASSES OF RINGS 4.1. Classes of Rings class operator A-closed Example 1: product Example 2:

4. CLASSES OF RINGS 4.1. Classes of Rings Normally we associate, with any property, a set of objects that satisfy that property. But problems can arise when we allow sets to be elements of larger sets

### NON SINGULAR MATRICES. DEFINITION. (Non singular matrix) An n n A is called non singular or invertible if there exists an n n matrix B such that

NON SINGULAR MATRICES DEFINITION. (Non singular matrix) An n n A is called non singular or invertible if there exists an n n matrix B such that AB = I n = BA. Any matrix B with the above property is called

### Definition: Group A group is a set G together with a binary operation on G, satisfying the following axioms: a (b c) = (a b) c.

Algebraic Structures Abstract algebra is the study of algebraic structures. Such a structure consists of a set together with one or more binary operations, which are required to satisfy certain axioms.

### Geometric Transformations

Geometric Transformations Definitions Def: f is a mapping (function) of a set A into a set B if for every element a of A there exists a unique element b of B that is paired with a; this pairing is denoted

### So let us begin our quest to find the holy grail of real analysis.

1 Section 5.2 The Complete Ordered Field: Purpose of Section We present an axiomatic description of the real numbers as a complete ordered field. The axioms which describe the arithmetic of the real numbers

### Lecture 6. Inverse of Matrix

Lecture 6 Inverse of Matrix Recall that any linear system can be written as a matrix equation In one dimension case, ie, A is 1 1, then can be easily solved as A x b Ax b x b A 1 A b A 1 b provided that

### Matrix Algebra. Some Basic Matrix Laws. Before reading the text or the following notes glance at the following list of basic matrix algebra laws.

Matrix Algebra A. Doerr Before reading the text or the following notes glance at the following list of basic matrix algebra laws. Some Basic Matrix Laws Assume the orders of the matrices are such that

### Practice Problems for First Test

Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the human mind will never penetrate.-

### CONSEQUENCES OF THE SYLOW THEOREMS

CONSEQUENCES OF THE SYLOW THEOREMS KEITH CONRAD For a group theorist, Sylow s Theorem is such a basic tool, and so fundamental, that it is used almost without thinking, like breathing. Geoff Robinson 1.

### ON GENERALIZED RELATIVE COMMUTATIVITY DEGREE OF A FINITE GROUP. A. K. Das and R. K. Nath

International Electronic Journal of Algebra Volume 7 (2010) 140-151 ON GENERALIZED RELATIVE COMMUTATIVITY DEGREE OF A FINITE GROUP A. K. Das and R. K. Nath Received: 12 October 2009; Revised: 15 December

### 4. Matrix inverses. left and right inverse. linear independence. nonsingular matrices. matrices with linearly independent columns

L. Vandenberghe EE133A (Spring 2016) 4. Matrix inverses left and right inverse linear independence nonsingular matrices matrices with linearly independent columns matrices with linearly independent rows

### a = bq + r where 0 r < b.

Lecture 5: Euclid s algorithm Introduction The fundamental arithmetic operations are addition, subtraction, multiplication and division. But there is a fifth operation which I would argue is just as fundamental

### Problem Set 5. AABB = 11k = (10 + 1)k = (10 + 1)XY Z = XY Z0 + XY Z XYZ0 + XYZ AABB

Problem Set 5 1. (a) Four-digit number S = aabb is a square. Find it; (hint: 11 is a factor of S) (b) If n is a sum of two square, so is 2n. (Frank) Solution: (a) Since (A + B) (A + B) = 0, and 11 0, 11

### CHAPTER 3 Numbers and Numeral Systems

CHAPTER 3 Numbers and Numeral Systems Numbers play an important role in almost all areas of mathematics, not least in calculus. Virtually all calculus books contain a thorough description of the natural,