# Group Theory. Chapter 1

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Chapter 1 Group Theory Most lectures on group theory actually start with the definition of what is a group. It may be worth though spending a few lines to mention how mathematicians came up with such a concept. Around 1770, Lagrange initiated the study of permutations in connection with the study of the solution of equations. He was interested in understanding solutions of polynomials in several variables, and got this idea to study the behaviour of polynomials when their roots are permuted. This led to what we now call Lagrange s Theorem, though it was stated as [5] If a function f(x 1,...,x n ) of n variables is acted on by all n! possible permutations of the variables and these permuted functions take on only r values, then r is a divisior of n!. It is Galois ( ) who is considered by many as the founder of group theory. He was the first to use the term group in a technical sense, though to him it meant a collection of permutations closed under multiplication. Galois theory will be discussed much later in these notes. Galois was also motivated by the solvability of polynomial equations of degree n. From 1815 to 1844, Cauchy started to look at permutations as an autonomous subject, and introduced the concept of permutations generated by certain elements, as well as several notations still used today, such as the cyclic notation for permutations, the product of permutations, or the identity permutation. He proved what we call today Cauchy s Theorem, namely that if p is prime divisor of the cardinality of the group, then there exists a subgroup of cardinality p. In 1870, Jordan gathered all the applications of permutations he could find, from algebraic geometry, number theory, function theory, and gave a unified presentation (including the work of Cauchy and Galois). Jordan made explicit the notions of homomorphism, isomorphism (still for permutation groups), he introduced solvable groups, and proved that the indices in two composition series are the same (now called Jordan-Hölder Theorem). He also gave a proof that the alternating group A n is simple for n > 4. In 1870, while working on number theory (more precisely, in generalizing 5

2 6 CHAPTER 1. GROUP THEORY Kummer s work on cyclotomic fields to arbitrary fields), Kronecker described in oneofhispapersafinitesetofarbitraryelementsonwhichhedefinedanabstract operation on them which satisfy certain laws, laws which now correspond to axioms for finite abelian groups. He used this definition to work with ideal classes. He also proved several results now known as theorems on abelian groups. Kronecker did not connect his definition with permutation groups, which was done in 1879 by Frobenius and Stickelberger. Apart permutation groups and number theory, a third occurence of group theory which is worth mentioning arose from geometry, and the work of Klein (we now use the term Klein group for one of the groups of order 4), and Lie, who studied transformation groups, that is transformations of geometric objects. The work by Lie is now a topic of study in itself, but Lie theory is beyond the scope of these notes. The abstract point of view in group theory emerged slowly. It took something like one hundred years from Lagrange s work of 1770 for the abstract group concept to evolve. This was done by abstracting what was in commun to permutation groups, abelian groups, transformation groups... In 1854, Cayley gave the modern definition of group for the first time: A set of symbols all of them different, and such that the product of any two of them (no matter in what order), or the product of any one of them into itself, belongs to the set, is said to be a group. These symbols are not in general convertible [commutative], but are associative. Let us start from there. 1.1 Groups and subgroups We start by introducing the object that will interest us for the whole chapter. Definition 1.1. A group is a non-empty set G on which there is a binary operation (a,b) ab such that if a and b belong to G then ab is also in G (closure), a(bc) = (ab)c for all a,b,c in G (associativity), there is an element 1 G such that a1 = 1a = a for all a G (identity), if a G, then there is an element a 1 G such that aa 1 = a 1 a = 1 (inverse). One can easily check that this implies the unicity of the identity and of the inverse. A group G is called abelian if the binary operation is commutative, i.e., ab = ba for all a,b G. Remark. There are two standard notations for the binary group operation: either the additive notation, that is (a,b) a + b in which case the identity is

3 1.1. GROUPS AND SUBGROUPS 7 denoted by 0, or the multiplicative notation, that is (a,b) ab for which the identity is denoted by 1. Examples Z with the addition and 0 as identity is an abelian group. 2. Z with the multiplication is not a group since there are elements which are not invertible in Z. 3. The set of n n invertible matrices with real coefficients is a group for the matrix product and identity the matrix I n. It is denoted by GL n (R) and called the general linear group. It is not abelian for n 2. The above examples are the easiest groups to think of. The theory of algebra however contains many examples of famous groups that one may discover, once equipped with more tools (for example, the Lie groups, the Brauer group, the Witt group, the Weyl group, the Picard group,...to name a few). Definition 1.2. The order of a group G, denoted by G, is the cardinality of G, that is the number of elements in G. We have only seen infinite groups so far. Let us look at some examples of finite groups. Examples The trivial group G = {0} may not be the most exciting group to look at, but still it is the only group of order The group G = {0,1,2,...,n 1} of integers modulo n is a group of order n. It is sometimes denoted by Z n (this should not be confused with p-adic integers though!). Definition 1.3. A subgroup H of a group G is a non-empty subset of G that forms a group under the binary operation of G. Examples If we consider the group G = Z 4 = {0,1,2,3} of integers modulo 4, H = {0,2} is a subgroup of G. 2. The set of n n matrices with real coefficients and determinant of 1 is a subgroup of GL n (R), denoted by SL n (R) and called the special linear group. At this point, in order to claim that the above examples are actually subgroups, one has to actually check the definition. The proposition below gives an easier criterion to decide whether a subset of a group G is actually a subgroup. Proposition 1.1. Let G be a group. Let H be a non-empty subset of G. The following are equivalent: 1. H is a subgroup of G. 2. (a) x,y H implies xy H for all x,y. (b) x H implies x 1 H.

4 8 CHAPTER 1. GROUP THEORY 3. x,y H implies xy 1 H for all x,y. Proof. We prove that This part is clear from the definition of subgroup Since H is non-empty, let x H. By assumption of 3., we have that xx 1 = 1 H and that 1x 1 H thus x is invertible in H. We now know that for x,y H, x and y 1 are in H, thus x(y 1 ) 1 = xy is in H To prove this direction, we need to check the definition of group. Since closure and existence of an inverse are true by assumption of 2., and that associativity follows from the associativity in G, we are left with the existence of an identity. Now, if x H, then x 1 H by assumption of 2., and thus xx 1 = 1 H again by assumption of 2., which completes the proof. We will often use the last equivalence to check that a subset of a group G is a subgroup. Now that we have these structures of groups and subgroups, let us introduce a map that allows to go from one group to another and that respects the respective group operations. Definition 1.4. Given two groups G and H, a group homomorphism is a map f : G H such that f(xy) = f(x)f(y) for all x,y G. Note that this definition immediately implies that the identity 1 G of G is mapped to the identity 1 H of H. The same is true for the inverse, that is f(x 1 ) = f(x) 1. Example 1.4. The map exp : (R,+) (R, ), x exp(x) is a group homomorphism. Definition 1.5. Two groups G and H are isomorphic if there is a group homomorphism f : G H which is also a bijection. Roughly speaking, isomorphic groups are essentially the same. Example 1.5. If we consider again the group G = Z 4 = {0,1,2,3} of integers modulo 4 with subgroup H = {0,2}, we have that H is isomorphic to Z 2, the group of integers modulo 2. A crucial definition is the definition of the order of a group element. Definition 1.6. The order of an element a G is the least positive integer n such that a n = 1. If no such integer exists, the order of a is infinite. We denote it by a.

5 1.2. CYCLIC GROUPS 9 Note that the critical part of this definition is that the order is the least positive integer with the given property. The terminology order is used both for groups and group elements, but it is usually clear from the context which one is considered. 1.2 Cyclic groups Let us now introduce a first family of groups, the cyclic groups. Definition 1.7. A group G is cyclic if it is generated by a single element, which we denote by G = a. We may denote by C n a cyclic group of n elements. Example 1.6. A finite cyclic group generated by a is necessarily abelian, and can be written (multiplicatively) {1,a,a 2,...,a n 1 } with a n = 1 or (additively) {0,a,2a,...,(n 1)a} with na = 0. A finite cyclic group with n elements is isomorphic to the additive group Z n of integers modulo n. Example 1.7. An nth root of unity is a complex number z which satisfies the equation z n = 1 for some positive integer n. Let ζ n = e 2iπ/n be an nth root of unity. All the nth roots of unity form a group under multiplication. It is a cyclic group, generated by ζ n, which is called a primitive root of unity. The term primitive exactly refers to being a generator of the cyclic group, namely, an nth root of unity is primitive when there is no positive integer k smaller than n such that ζ k n = 1. Here are some properties of cyclic groups and its generators. Proposition 1.2. If G is a cyclic group of order n generated by a, the following conditions are equivalent: 1. a k = n. 2. k and n are relatively prime. 3. k has an inverse modulo n, that is there exists an integer s such that ks 1 modulo n. Proof. Before starting the proof, recall that since a generates G of order n, we have that the order of a is n and in particular a n = 1. The fact that a k = n means in words that the order of a k is also n, that is, a k is also a generator of G. We first prove that 1. 2., while follows from Bezout identity.

6 10 CHAPTER 1. GROUP THEORY Suppose by contradiction that k and n are not relatively prime, that is, there exists s > 1 such that s k and s n. Thus n = ms and k = sr for some m,r 1 and we have (a k ) m = a srm = a nr = 1. Now since s > 1, m < n, which contradicts that n is the order of a k Suppose that the order of a k is not n, then there exists m < n such that (a k ) m = 1 and thus n km since n is the order of a. If k and n were to be relatively prime, then n would divide m, which is a contradiction since m < n If k and n are relatively prime, then by Bezout identity, there exist r,s such that 1 = kr +ns and thus kr 1 modulo n If kr 1 modulo n then 1 = kr+ns for some s and the greatest common divisor of k and n must divide 1, which shows k and n are relatively prime. Corollary 1.3. The set of invertible elements modulo n forms a group under multiplication, whose order is the Euler function ϕ(n), which by definition counts the number of positive integers less than n that are relatively prime to n. Example 1.8. Consider the group Z 6 = {0,1,2,3,4,5}, the group Z 6 of invertible elements in Z 6 is Z 6 = {1,5}. We have that ϕ(6) = ϕ(2)ϕ(3) = Cosets and Lagrange s Theorem Definition 1.8. Let H be a subgroup of a group G. If g G, the right coset of H generated by g is Hg = {hg, h H} and similarly the left coset of H generated by g is gh = {gh, h H}. In additive notation, we get H +g (which usually implies that we deal with a commutative group where we do not need to distinguish left and right cosets). Example 1.9. If we consider the group Z 4 = {0,1,2,3} and its subgroup H = {0,2} which is isomorphic to Z 2, the cosets of H in G are 0+H = H, 1+H = {1,3}, 2+H = H, 3+H = {1,3}. Clearly 0+H = 2+H and 1+H = 3+H. We see in the above example that while an element of g G runs through all possible elements of the group G, some of the left cosets gh (or right cosets Hg) may be the same. It is easy to see when this exactly happens.

7 1.3. COSETS AND LAGRANGE S THEOREM 11 Lemma 1.4. We have that Ha = Hb if and only if ab 1 H for a,b G. Similarly, ah = bh if and only if a 1 b H for a,b G. Proof. If two right cosets are the same, that is Ha = Hb, since H is a subgroup, we have 1 H and a = hb for some h H, so ab 1 = h H. Conversely, if ab 1 = h H, then Ha = Hhb = Hb, again since H is a subgroup. While one may be tempted to define a coset with a subset of G which is not a subgroup, we see that the above characterization really relies on the fact that H is actually a subgroup. Example Itisthusnosurprisethatintheaboveexamplewehave0+H = 2 + H and 1 + H = 3 + H, since we have modulo 4 that H and H. Saying that two elements a,b G generate the same coset is actually an equivalence relation in the following sense. We say that a is equivalent to b if and only if ab 1 H, and this relation satisfies the three properties of an equivalence relation: reflexivity: aa 1 = 1 H. symmetry: if ab 1 H then (ab 1 ) 1 = ba 1 H. transitivity: if ab 1 H and bc 1 H then (ab 1 )(bc 1 ) = ac 1 H. The equivalence class of a is the set of elements in G which are equivalent to a, namely {b, ab 1 H}. Since ab 1 H (ab 1 ) 1 = ba 1 H b Ha, we further have that {b, ab 1 H} = Ha, and a coset is actually an equivalence class. Example Let us get back to our example with the group Z 4 = {0,1,2,3} and its subgroup H = {0,2}. We compute the first coset 0+H = H, and thus we now know that the equivalence class of 0 is H, and thus there is no need to compute the coset generated by 2, since it will give the same coset. We then compute the coset 1 + H = {1,3} and again there is no need to compute the one of 3 since it is already in the coset of 1. We thus get 2 cosets, and clearly they partition Z 4 : Z 4 = {0,2} {1,3} = H (1+H). It is important to notice that the right (resp. left) cosets partition the group G (that the union of all cosets is G is clear since we run through all elements of G and H contains 1, and it is easy to see that if x Ha and x Hb then Ha = Hb).

8 12 CHAPTER 1. GROUP THEORY Example Consider R as an additive group with subgroup Z. Every real number up to addition by an integer looks like a number in [0,1). Thus and the cosets of Z partition R. R = 0 x<1 (x+z), Furthermore, since the map h ha, h H, is a one-to-one correspondence, each coset has H elements. Definition 1.9. The index of a subgroup H in G is the number of right (left) cosets. It is a positive number or and is denoted by [G : H]. If we think of a group G as being partitioned by cosets of a subgroup H, then the index of H tells how many times we have to translate H to cover the whole group. Example In Example 1.12, the index [R : Z] is infinite, since there are infinitely many cosets of Z in R. Theorem 1.5. (Lagrange s Theorem). If H is a subgroup of G, then G = H [G : H]. In particular, if G is finite then H divides G and [G : H] = G / H. Proof. Let us start by recalling that the left cosets of H forms a partition of G, that is G = gh, where g runs through a set of representatives (one for each coset). Let us look at the cardinality of G: G = gh = gh since we have a disjoint union of cosets, and the sum is again over the set of representatives. Now gh = H since we have already noted that each coset contains H elements. We then conclude that G = H = [G : H] H. Example Consider G = Z, H = 3Z, then [G : H] = 3. Of course, Lagrange did not prove Lagrange s theorem! The modern way of defining groups did not exist yet at his time. Lagrange was interested in polynomial equations, and in understanding the existence and nature of the roots (does every equation has a root? how many roots?...). What he actually proved was that if a polynomial in n variables has its variables permuted in all n! ways, the number of different polynomials that are obtained is always a

9 1.3. COSETS AND LAGRANGE S THEOREM 13 Figure 1.1: Joseph-Louis Lagrange ( ) factor of n!. Since all the permutations of n elements are actually a group, the number of such polynomials is actually the index in the group of permutations of n elements of the subgroup H of permutations which preserve the polynomial. So the size of H divides n!, which is exactly the number of all permutations of n elements. This is indeed a particular case of what we call now Lagrange s Theorem. Corollary Let G be a finite group. If a G, then a divides G. In particular, a G = If G has prime order, then G is cyclic. Proof. 1. If a G has order say m, then the subgroup H = {1,a,...,a m 1 } is a cyclic subgroup of G with order H = m. Thus m divides G by the theorem. 2. Since G is prime, we may take a 1 in G, and since the order of a has to divide G, we have a = G. Thus the cyclic group generated by a coincides with G. Example Using Lagrange s Theorem and its corollaries, we can already determine the groups of order from 1 to 5, up to isomorphism (see Table 1.1). If G is prime, we now know that G is cyclic. Let us look at the case where G is of order 4. Let g G. We know that the order of g is either 1,2 or 4. If the order of g is 1, this is the identity. If G contains an element g of order 4, then that means that g generates the whole group, thus G is cyclic. If now G does not contain an element of order 4, then

10 14 CHAPTER 1. GROUP THEORY G G 1 {1} 2 C 2 3 C 3 4 C 4, C 2 C 2 5 C 5 Table 1.1: Groups of order from 1 to 5. C n denotes the cyclic group of order n. apart the identity, all the elements have order 2. From there, it is easy to obtain a multiplication table for G, and see that it coincides with the one of the group Z 2 Z 2 = {(x,y) x,y Z 2 } with binary operation (x,y)+(x,y ) = (x+x,y+y ). This group is called the Klein group, and it has several interpretations, for example, it is the group of isometries fixing a rectangle. Remark. The above example also shows that the converse of Lagrange s Theorem is not true. If we take the group G = C 2 C 2, then 4 divides the order of G, however there is no element of order 4 in G. Once Lagrange s Theorem and its corollaries are proven, we can easily deduce Euler s and Fermat s Theorem. Theorem 1.7. (Euler s Theorem). If a and n are relatively prime positive integers, with n 2, then a ϕ(n) 1 mod n. Proof. Since a and n are relatively prime, we know from Proposition 1.2 that a has an inverse modulo n, and by its corollary that the group of invertible elements has order ϕ(n). Thus by Lagrange s Theorem first corollary. a ϕ(n) 1 mod n Corollary 1.8. (Fermat s Little Theorem). If p is a prime and a is a positive integer not divisible by p, then a p 1 1 mod p. This is particular case of Euler s Theorem when n is a prime, since then ϕ(n) = p 1.

11 1.4. NORMAL SUBGROUPS AND QUOTIENT GROUP Normal subgroups and quotient group Given a group G and a subgroup H, we have seen how to define the cosets of H, and thanks to Lagrange s Theorem, we already know that the number of cosets [G : H] is related to the order of H and G by G = H [G : H]. A priori, the set of cosets of H has no structure. We are now interested in a criterion on H to give the set of its cosets a structure of group. In what follows, we may write H G for H is a subgroup of G. Definition Let G be a group and H G. We say that H is a normal subgroup of G, or that H is normal in G, if we have chc 1 = H, for all c G. We denote it H G, or H G when we want to emphasize that H is a proper subgroup of G. The condition for a subgroup to be normal can be stated in many slightly different ways. Lemma 1.9. Let H G. The following are equivalent: 1. chc 1 H for all c G. 2. chc 1 = H for all c G, that is ch = Hc for all c G. 3. Every left coset of H in G is also a right coset (and vice-versa, every right coset of H in G is also a left coset). Proof. Clearly 2. implies 1., now chc 1 H for all c G if and only if ch Hc. Let x Hc, that is x = hc for some h H, so that x = (cc 1 )hc = c(c 1 hc) = ch for some h H since chc 1 H for all c and thus in particular for c 1. This shows that Hc is included in ch or equivalently that H chc 1. Also 2. clearly implies 3. Now suppose that ch = Hd. This means that c belongs to Hd by assumption and to Hc by definition, which means that Hd = Hc. Example Let GL n (R) be the group of n n real invertible matrices, and let SL n (R) be the subgroup formed by matrices whose determinant is 1. Let us see that SL n (R) GL n (R). For that, we have to check that ABA 1 SL n (R) for all B SL n (R) and A GL n (R). This is clearly true since det(aba 1 ) = det(b) = 1. Proposition If H is normal in G, then the cosets of H form a group.

12 16 CHAPTER 1. GROUP THEORY Proof. Letusfirstdefineabinaryoperationonthecosets: (ah,bh) (ah)(bh) = {(ah)(bh ), ah ah, bh bh}. We need to check that the definition of group is satisfied. closure. This is the part which asks a little bit of work. Since ch = Hc for all c G, then (ah)(bh) = a(hb)h = a(bh)h = abhh = abh. Note that this product does not depend on the choice of representatives. Associativity comes from G being associative. The identity is given by the coset 1H = H. The inverse of the coset ah is a 1 H. Definition The group of cosets of a normal subgroup N of G is called the quotient group of G by N. It is denoted by G/N. Let us finish this section by discussing the connection between normal subgroups and homomorphisms. The first normal subgroup of interest will be the kernel of a group homomorphism. Recall that if f : G H is a group homomorphism, the kernel of f is defined by Ker(f) = {a G, f(a) = 1}. It is easy to see that Ker(f) is a normal subgroup of G, since f(aba 1 ) = f(a)f(b)f(a) 1 = f(a)f(a) 1 = 1 for all b Ker(f) and all a G. The converse is more interesting. Proposition Let G be a group. Every normal subgroup of G is the kernel of a homomorphism. Proof. Suppose that N G and consider the map π : G G/N, a an. To prove the result, we have to show that π is a group homomorphism whose kernel is N. First note that π is indeed a map from group to group since G/N is a group by assuming that N is normal. Then we have that π(ab) = abn = (an)(bn) = π(a)π(b) where the second equality comes from the group structure of G/N. Finally Ker(π) = {a G π(a) = N} = {a G an = N} = N.

13 1.4. NORMAL SUBGROUPS AND QUOTIENT GROUP 17 Definition Let N G. The group homomorphism π : G G/N, a an is called the natural or canonical map or projection. Recall for further usage that for f a group homomorphism, we have the following characterization of injectivity: a homomorphism f is injective if and only if its kernel is trivial (that is, contains only the identity element). Indeed, if f is injective, then Ker(f) = {a, f(a) = 1} = {1} since f(1) = 1. Conversely, if Ker(f) = {1} and we assume that f(a) = f(b), then f(ab 1 ) = f(a)f(b) 1 = f(a)f(a) 1 = 1 and ab 1 = 1 implying that a = b and thus f is injective. Terminology. monomorphism=injective homomorphism epimorphism=surjective homomorphism isomorphism=bijective homomorphism endomorphism=homomorphism of a group to itself automorphism=isomorphism of a group with itself We have looked so far at the particular subgroup of G which is its kernel. The proposition below describes more generally subgroups of G and H. Proposition Let f : G H be a homomorphism. 1. If K is a subgroup of G, then f(k) is a subgroup of H. If f is an epimorphism and K is normal, then f(k) is normal. 2. If K is a subgroup of H, then f 1 (K) = {x G, f(x) K} is a subgroup of G. If K is normal, so is f 1 (K). Proof. 1. To prove that f(k) is a subgroup of H, it is enough to show that f(a)f(b) 1 f(k) by Proposition 1.1, which is clear from f(a)f(b) 1 = f(ab 1 ) f(k). If K is normal, we have to show that cf(k)c 1 = f(k) for all c H. Since f is an epimorphism, there exists d G such that f(d) = c, so that cf(k)c 1 = f(d)f(k)f(d) 1 = f(dkd 1 ) = f(k) using that K is normal.

14 18 CHAPTER 1. GROUP THEORY 2. As before, to prove that f 1 (K) is a subgroup of G, it is enough to showing that ab 1 f 1 (K) for a,b f 1 (K), which is equivalent to show that f(ab 1 ) K. This is now true since f(ab 1 ) = f(a)f(b) 1 with a,b f 1 (K) and K a subgroup. For the second claim, we have to show that or equivalently For c G and a f 1 (K), then since K is normal. cf 1 (K)c 1 = f 1 (K) f(cf 1 (K)c 1 ) = K, c G. f(cac 1 ) = f(c)f(a)f(c) 1 K 1.5 The isomorphism theorems This section presents different isomorphism theorems which are important tools for proving further results. The first isomorphism theorem, that will be the second theorem to be proven after the factor theorem, is easier to motivate, since it will help us in computing quotient groups. But let us first start with the so-called factor theorem. Assume that we have a group G which contains a normal subgroup N, another group H, and f : G H a group homomorphism. Let π be the canonical projection (see Definition 1.12) from G to the quotient group G/N: G f H π G/N We would like to find a homomorphism f : G/N H that makes the diagram commute, namely f(a) = f(π(a)) for all a G. Theorem (Factor Theorem). Any homomorphism f whose kernel K contains N can be factored through G/N. In other words, there is a unique homomorphism f : G/N H such that f π = f. Furthermore f 1. f is an epimorphism if and only if f is. 2. f is a monomorphism if and only if K = N.

15 1.5. THE ISOMORPHISM THEOREMS f is an isomorphism if and only if f is an epimorphism and K = N. Proof. Unicity. Letusstartbyprovingthatifthereexists f suchthat f π = f, then it is unique. Let f be another homomorphism such that f π = f. We thus have that ( f π)(a) = ( f π)(a) = f(a) for all a G, that is f(an) = f(an) = f(a). This tells us that for all bn G/N for which there exists an element b in G such that π(b) = bn, then its image by either f or f is determined by f(b). This shows that f = f by surjectivity of π. Existence. Let an G/N such that π(a) = an for a G. We define f(an) = f(a). This is the most natural way to do it, however, we need to make sure that this is indeed well-defined, in the sense that it should not depend on the choice of the representative taken in the coset. Let us thus take another representative, say b an. Since a and b are in the same coset, they satisfy a 1 b N K, where K = Ker(f) by assumption. Since a 1 b K, we have f(a 1 b) = 1 and thus f(a) = f(b). Nowthat f iswelldefined, letuscheckthisisindeedagrouphomomorphism. First note that G/N is indeed a group since N G. Then, we have f(anbn) = f(abn) = f(ab) = f(a)f(b) = f(an) f(bn) and f is a homomorphism. 1. The fact that f is an epimorphism if and only if f is comes from the fact that both maps have the same image. 2. First note that the statement f is a monomorphism if and only if K = N makes sense since K = Ker(f) is indeed a normal subgroup, as proved earlier. To show that f is a monomorphism is equivalent to show that Ker( f) is trivial. By definition, we have Ker( f) = {an G/N, f(an) = 1} = {an G/N, f(π(a)) = f(a) = 1} = {an G/N, a K = Ker(f)}. So the kernel of f is exactly those cosets of the form an with a K, but for the kernel to be trivial, we need it to be equal to N, that is we need K = N. 3. This is just a combination of the first two parts.

16 20 CHAPTER 1. GROUP THEORY We are now ready to state the first isomorphism theorem. Theorem (1st Isomorphism Theorem). If f : G H is a homomorphism with kernel K, then the image of f is isomorphic to G/K: Im(f) G/Ker(f). Proof. We know from the Factor Theorem that f : G/Ker(f) H is an isomorphism if and only if f is an epimorphism, and clearly f is an epimorphism on its image, which concludes the proof. Example We have seen in Example 1.16 that SL n (R) GL n (R). Consider the map det : GL n (R) (R, ), which is a group homomorphism. We have that Ker(det) = SL n (R). The 1st Isomorphism Theorem tells that Im(det) GL n (R)/SL n (R). It is clear that det is surjective, since for all a R, one can take the diagonal matrix with all entries at 1, but one which is a. Thus we conclude that R GL n (R)/SL n (R). The 1st Isomorphism Theorem can be nicely illustrated in terms of exact sequences. Definition Let F,G,H,I,... be groups, and let f,g,h,... be group homomorphisms. Consider the following sequence: F f G g H h I We say that this sequence is exact in one point (say G) if Im(f) = Ker(g). A sequence is exact if it is exact in all points. A short exact sequence of groups is of the form 1 i F f G g H j 1 where i is the inclusion and j is the constant map 1. Proposition Let 1 i F f G g H j 1

17 1.5. THE ISOMORPHISM THEOREMS 21 be a short exact sequence of groups. Then Im(f) is normal in G and we have a group isomorphism G/Im(f) H, or equivalently G/Ker(g) H. Proof. Since the sequence is exact, we have that Im(f) = Ker(g) thus Im(f) is a normal subgroup of G. By the first Isomorphism Theorem, we have that since Im(g) = Ker(j) = H. G/Ker(g) Im(g) = H. The formulation in terms of exact sequences is useful to know, since it happens very often in the literature that an exact sequence is given exactly to be able to compute such quotient groups. Let us state the second and third isomorphism theorem. Theorem (2nd Isomorphism Theorem). If H and N are subgroups of G, with N normal in G, then H/(H N) HN/N. There are many things to discuss about the statement of this theorem. First we need to check that HN is indeed a subgroup of G. To show that, notice that HN = NH since N is a normal subgroup of G. This implies that for hn HN, its inverse (hn) 1 = n 1 h 1 G actually lives in HN, and so does the product (hn)(h n ) = h(nh )n. Note that by writing HN/N, we insist on the fact that there is no reason for N to be a subgroup of H. On the other hand, N is a normal subgroup of HN, since for all hn HN, we have since N is normal in G. hnnn 1 h 1 = hnh 1 N We now know that the right hand side of the isomorphism is a quotient group. In order to see that so is the left hand side, we need to show that H N is a normal subgroup of H. This comes by noticing that H N is the kernel of the canonical map π : G G/N restricted to H. Now that all these remarks have been done, it is not difficult to see that the 2nd Isomorphism Theorem follows from the 1st Isomorphism Theorem, as does the 3rd Isomorphism Theorem. Theorem (3rd Isomorphism Theorem). If N and H are normal subgroups of G, with N contained in H, then G/H (G/N)/(H/N).

18 22 CHAPTER 1. GROUP THEORY 1.6 Direct and semi-direct products So far, we have seen how given a group G, we can get smaller groups, such as subgroups of G or quotient groups. We will now do the other way round, that is, starting with a collection of groups, we want to build larger new groups. Let us start with two groups H and K, and let G = H K be the cartesian product of H and K, that is G = {(h,k), h H, k K}. We define a binary operation on this set by doing componentwise multiplication (or addition if the binary operations of H and K are denoted additively) on G: (h 1,k 1 )(h 2,k 2 ) = (h 1 h 2,k 1 k 2 ) H K. Clearly G is closed under multiplication, its operation is associative (since both operations on H and K are), it has an identity element given by 1 G = (1 H,1 K ) and the inverse of (h,k) is (h 1,k 1 ). In summary, G is a group. Definition Let H, K be two groups. The group G = H K with binary operation defined componentwise as described above is called the external direct product of H and K. Examples Let Z 2 be the group of integers modulo 2. We can build a direct product of Z 2 with itself, namely Z 2 Z 2 with additive law componentwise. This is actually the Klein group, also written C 2 C 2. This group is not isomorphic to Z 4! 2. Let Z 2 be the group of integers modulo 2, and Z 3 be the group of integers modulo 3. We can build a direct product of Z 2 and Z 3, namely Z 2 Z 3 with additive law componentwise. This group is actually isomorphic to Z 6! 3. The group(r, +) (R, +) with componentwise addition is a direct product. Note that G contains isomorphic copies H and K of respectively H and K, given by H = {(h,1 K ), h H}, K = {(1H,k), k K}, which furthermore are normal subgroups of G. Let us for example see that H is normal in G. By definition, we need to check that Let (h,1 K ) H, we compute that (h,k) H(h 1,k 1 ) H, (h,k) G. (h,k)(h,1 k )(h 1,k 1 ) = (hh h 1,1 k ) H, since hh h 1 H. The same computation holds for K. If we gather what we know about G, H and K, we get that

19 1.6. DIRECT AND SEMI-DIRECT PRODUCTS 23 by definition, G = H K and H K = {1 G }, by what we have just proved, H and K are two normal subgroups of G. This motivates the following definition. Definition If a group G contains normal subgroups H and K such that G = HK and H K = {1 G }, we say that G is the internal direct product of H and K. Examples Consider the Klein group Z 2 Z 2, it contains the two subgroups H = {(h,0), h Z 2 } and K = {(0,k), k Z 2 }. We have that both H and K are normal, because the Klein group is commutative. We also have that H K = {(0,0)}, so the Klein group is indeed an internal direct product. On the other hand, Z 4 only contains as subgroup H = {0,2}, so it is not an internal direct product! 2. ConsiderthegroupZ 2 Z 3, itcontainsthetwosubgroupsh = {(h,0), h Z 2 } and K = {(0,k), k Z 3 }. We have that both H and K are normal, because the group is commutative. We also have that H K = {(0,0)}, so this group is indeed an internal direct product. Also Z 6 contains the two subgroups H = {0,3} Z 2 and K = {0,2,4} Z 3. We have that both H and K are normal, because the group is commutative. We also have that H K = {0}, so this group is indeed an internal direct product, namely the internal product of Z 2 and Z 3. This is in fact showing that Z 6 Z 2 Z 3. The next result makes explicit the connection between internal and external products. Proposition If G is the internal direct product of H and K, then G is isomorphic to the external direct product H K. Proof. To show that G is isomorphic to H K, we define the following map f : H K G, f(h,k) = hk. First remark that if h H and k K, then hk = kh. Indeed, we have using that both K and H are normal that implying that (hkh 1 )k 1 K, h(kh 1 k 1 ) H hkh 1 k 1 K H = {1}. We are now ready to prove that f is a group isomorphism. 1. This is a group homomorphism since f((h,k)(h,k )) = f(hh,kk ) = h(h k)k = h(kh )k = f(h,k)f(h,k ).

20 24 CHAPTER 1. GROUP THEORY 2. The map f is injective. This can be seen by checking that its kernel is trivial. Indeed, if f(h,k) = 1 then hk = 1 h = k 1 h K h H K = {1}. We have then that h = k = 1 which proves that the kernel is {(1,1)}. 3. The map f is surjective since by definition G = HK. Note that the definitions of external and internal product are surely not restricted to two groups. One can in general define them for n groups H 1,...,H n. Namely Definition If H 1,...,H n are arbitrary groups, the external direct product of H 1,...,H n is the cartesian product G = H 1 H 2 H n with componentwise multiplication. If G contains normal subgroups H 1,...,H n such that G = H 1 H n and each g can be represented as h 1 h n uniquely, we say that G is the internal direct product of H 1,...,H n. We can see a slight difference in the definition of internal product, since in the case of two subgroups, the condition given was not that each g can be represented uniquely as h 1 h 2, but instead that the intersection of the two subgroups is {1}. The next proposition shows the connection between these two points of view. Proposition Suppose that G = H 1 H n where each H i is a normal subgroup of G. The following conditions are equivalent. 1. G is the internal direct product of the H i. 2. H 1 H 2 H i 1 H i = {1}, for all i = 1,...,n. Proof. Let us prove Let us assume that G is the internal direct product of the H i, which means that every element in G can be written uniquely as a product of elements in H i. Now let us take g H 1 H 2 H i 1 H i = {1}. We have that g H 1 H 2 H i 1, which is uniquely written as g = h 1 h 2 h i 1 1 Hi 1 Hn, h j H j. On the other hand, g H i thus g = 1 H1 1 Hi 1 g and by unicity of the representation, we have h j = 1 for all j and g = Conversely, let us assume that g G can be written either g = h 1 h 2 h n, h j H j,

### 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

### 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

### 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

### 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

### 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

### Revision of ring theory

CHAPTER 1 Revision of ring theory 1.1. Basic definitions and examples In this chapter we will revise and extend some of the results on rings that you have studied on previous courses. A ring is an algebraic

### 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

### 4. FIRST STEPS IN THE THEORY 4.1. A

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

### 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

### 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

### 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

GROUP ACTIONS KEITH CONRAD 1. Introduction The symmetric groups S n, alternating groups A n, and (for n 3) dihedral groups D n behave, by their very definition, as permutations on certain sets. The groups

### 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)

### 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...........................................

### 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

### 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

### D-MATH Algebra I HS 2013 Prof. Brent Doran. Solution 5

D-MATH Algebra I HS 2013 Prof. Brent Doran Solution 5 Dihedral groups, permutation groups, discrete subgroups of M 2, group actions 1. Write an explicit embedding of the dihedral group D n into the symmetric

### 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

### 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.

### Sets and functions. {x R : x > 0}.

Sets and functions 1 Sets The language of sets and functions pervades mathematics, and most of the important operations in mathematics turn out to be functions or to be expressible in terms of functions.

### ON GALOIS REALIZATIONS OF THE 2-COVERABLE SYMMETRIC AND ALTERNATING GROUPS

ON GALOIS REALIZATIONS OF THE 2-COVERABLE SYMMETRIC AND ALTERNATING GROUPS DANIEL RABAYEV AND JACK SONN Abstract. Let f(x) be a monic polynomial in Z[x] with no rational roots but with roots in Q p for

### 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

### Notes on Group Theory

Notes on Group Theory Mark Reeder March 7, 2014 Contents 1 Notation for sets and functions 4 2 Basic group theory 4 2.1 The definition of a group................................. 4 2.2 Group homomorphisms..................................

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### F1.3YE2/F1.3YK3 ALGEBRA AND ANALYSIS. Part 2: ALGEBRA. RINGS AND FIELDS

F1.3YE2/F1.3YK3 ALGEBRA AND ANALYSIS Part 2: ALGEBRA. RINGS AND FIELDS LECTURE NOTES AND EXERCISES Contents 1 Revision of Group Theory 3 1.1 Introduction................................. 3 1.2 Binary Operations.............................

### 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

### 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

### 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

### 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

### 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.

### ABSTRACT ALGEBRA. Romyar Sharifi

ABSTRACT ALGEBRA Romyar Sharifi Contents Introduction 7 Part 1. A First Course 11 Chapter 1. Set theory 13 1.1. Sets and functions 13 1.2. Relations 15 1.3. Binary operations 19 Chapter 2. Group theory

### Ideal Class Group and Units

Chapter 4 Ideal Class Group and Units We are now interested in understanding two aspects of ring of integers of number fields: how principal they are (that is, what is the proportion of principal ideals

### 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

### EXERCISES FOR THE COURSE MATH 570, FALL 2010

EXERCISES FOR THE COURSE MATH 570, FALL 2010 EYAL Z. GOREN (1) Let G be a group and H Z(G) a subgroup such that G/H is cyclic. Prove that G is abelian. Conclude that every group of order p 2 (p a prime

### NOTES ON GROUP THEORY

NOTES ON GROUP THEORY Abstract. These are the notes prepared for the course MTH 751 to be offered to the PhD students at IIT Kanpur. Contents 1. Binary Structure 2 2. Group Structure 5 3. Group Actions

### 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,

### 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

### Chapter 10. Abstract algebra

Chapter 10. Abstract algebra C.O.S. Sorzano Biomedical Engineering December 17, 2013 10. Abstract algebra December 17, 2013 1 / 62 Outline 10 Abstract algebra Sets Relations and functions Partitions and

### 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

### 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

### MT2002 Algebra Edmund F Robertson

MT2002 Algebra 2002-3 Edmund F Robertson September 3, 2002 2 Contents Contents... 2 About the course... 4 1. Introduction... 4 2. Groups definition and basic properties... 6 3. Modular arithmetic... 8

### 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

### 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

### 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

### Module MA3411: Abstract Algebra Galois Theory Appendix Michaelmas Term 2013

Module MA3411: Abstract Algebra Galois Theory Appendix Michaelmas Term 2013 D. R. Wilkins Copyright c David R. Wilkins 1997 2013 Contents A Cyclotomic Polynomials 79 A.1 Minimum Polynomials of Roots of

### 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,

### MA651 Topology. Lecture 6. Separation Axioms.

MA651 Topology. Lecture 6. Separation Axioms. This text is based on the following books: Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology by Nicolas Bourbaki Counterexamples

### 9. Quotient Groups Given a group G and a subgroup H, under what circumstances can we find a homomorphism φ: G G ', such that H is the kernel of φ?

9. Quotient Groups Given a group G and a subgroup H, under what circumstances can we find a homomorphism φ: G G ', such that H is the kernel of φ? Clearly a necessary condition is that H is normal in G.

### IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL. 1. Introduction

IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL R. DRNOVŠEK, T. KOŠIR Dedicated to Prof. Heydar Radjavi on the occasion of his seventieth birthday. Abstract. Let S be an irreducible

### 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

### = 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

### Mathematics for Computer Science/Software Engineering. Notes for the course MSM1F3 Dr. R. A. Wilson

Mathematics for Computer Science/Software Engineering Notes for the course MSM1F3 Dr. R. A. Wilson October 1996 Chapter 1 Logic Lecture no. 1. We introduce the concept of a proposition, which is a statement

### Review for Final Exam

Review for Final Exam Note: Warning, this is probably not exhaustive and probably does contain typos (which I d like to hear about), but represents a review of most of the material covered in Chapters

### 11 Ideals. 11.1 Revisiting Z

11 Ideals The presentation here is somewhat different than the text. In particular, the sections do not match up. We have seen issues with the failure of unique factorization already, e.g., Z[ 5] = O Q(

### INTRODUCTORY SET THEORY

M.Sc. program in mathematics INTRODUCTORY SET THEORY Katalin Károlyi Department of Applied Analysis, Eötvös Loránd University H-1088 Budapest, Múzeum krt. 6-8. CONTENTS 1. SETS Set, equal sets, subset,

### In mathematics you don t understand things. You just get used to them. (Attributed to John von Neumann)

Chapter 1 Sets and Functions We understand a set to be any collection M of certain distinct objects of our thought or intuition (called the elements of M) into a whole. (Georg Cantor, 1895) In mathematics

### COMMUTATIVITY DEGREES OF WREATH PRODUCTS OF FINITE ABELIAN GROUPS

Bull Austral Math Soc 77 (2008), 31 36 doi: 101017/S0004972708000038 COMMUTATIVITY DEGREES OF WREATH PRODUCTS OF FINITE ABELIAN GROUPS IGOR V EROVENKO and B SURY (Received 12 April 2007) Abstract We compute

### Group Theory: Basic Concepts

Group Theory: Basic Concepts Robert B. Griffiths Version of 9 Feb. 2009 References: EDM = Encyclopedic Dictionary of Mathematics, 2d English edition (MIT, 1987) HNG = T. W. Hungerford: Algebra (Springer-Verlag,

### 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

### 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

### CARDINALITY, COUNTABLE AND UNCOUNTABLE SETS PART ONE

CARDINALITY, COUNTABLE AND UNCOUNTABLE SETS PART ONE With the notion of bijection at hand, it is easy to formalize the idea that two finite sets have the same number of elements: we just need to verify

### Some Basic Techniques of Group Theory

Chapter 5 Some Basic Techniques of Group Theory 5.1 Groups Acting on Sets In this chapter we are going to analyze and classify groups, and, if possible, break down complicated groups into simpler components.

### Sets and Cardinality Notes for C. F. Miller

Sets and Cardinality Notes for 620-111 C. F. Miller Semester 1, 2000 Abstract These lecture notes were compiled in the Department of Mathematics and Statistics in the University of Melbourne for the use

### 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

### 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

### Proof. Right multiplication of a permutation by a transposition of neighbors either creates a new inversion or kills an existing one.

GROUPS AROUND US Pavel Etingof Introduction These are notes of a mini-course of group theory for high school students that I gave in the Summer of 2009. This mini-course covers the most basic parts of

### Classical Analysis I

Classical Analysis I 1 Sets, relations, functions A set is considered to be a collection of objects. The objects of a set A are called elements of A. If x is an element of a set A, we write x A, and if

### 21 Homomorphisms and Normal Subgroups

Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan 21 Homomorphisms and Normal Subgroups Recall that an isomorphism is a function θ : G H such that θ is one-to-one, onto

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

### 2. Groups I. 1. Groups

2. Groups I 2.1 Groups 2.2 Subgroups, Lagrange s theorem 2.3 Homomorphisms, kernels, normal subgroups 2.4 Cyclic groups 2.5 Quotient groups 2.6 Groups acting on sets 2.7 The Sylow theorem 2.8 Trying to

### S(A) X α for all α Λ. Consequently, S(A) X, by the definition of intersection. Therefore, X is inductive.

MA 274: Exam 2 Study Guide (1) Know the precise definitions of the terms requested for your journal. (2) Review proofs by induction. (3) Be able to prove that something is or isn t an equivalence relation.

### 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

### it is easy to see that α = a

21. Polynomial rings Let us now turn out attention to determining the prime elements of a polynomial ring, where the coefficient ring is a field. We already know that such a polynomial ring is a UF. Therefore

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

### 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

### COMMUTATIVITY DEGREES OF WREATH PRODUCTS OF FINITE ABELIAN GROUPS

COMMUTATIVITY DEGREES OF WREATH PRODUCTS OF FINITE ABELIAN GROUPS IGOR V. EROVENKO AND B. SURY ABSTRACT. We compute commutativity degrees of wreath products A B of finite abelian groups A and B. When B

### 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

### 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,

### Finite Fields and Error-Correcting Codes

Lecture Notes in Mathematics Finite Fields and Error-Correcting Codes Karl-Gustav Andersson (Lund University) (version 1.013-16 September 2015) Translated from Swedish by Sigmundur Gudmundsson Contents

### Quadratic Equations in Finite Fields of Characteristic 2

Quadratic Equations in Finite Fields of Characteristic 2 Klaus Pommerening May 2000 english version February 2012 Quadratic equations over fields of characteristic 2 are solved by the well known quadratic

### Galois Theory III. 3.1. Splitting fields.

Galois Theory III. 3.1. Splitting fields. We know how to construct a field extension L of a given field K where a given irreducible polynomial P (X) K[X] has a root. We need a field extension of K where

### 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

University of Oslo MAT2 Project The Banach-Tarski Paradox Author: Fredrik Meyer Supervisor: Nadia S. Larsen Abstract In its weak form, the Banach-Tarski paradox states that for any ball in R, it is possible

### 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

### Introducing Functions

Functions 1 Introducing Functions A function f from a set A to a set B, written f : A B, is a relation f A B such that every element of A is related to one element of B; in logical notation 1. (a, b 1

### 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

### 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

### 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

### Factoring of Prime Ideals in Extensions

Chapter 4 Factoring of Prime Ideals in Extensions 4. Lifting of Prime Ideals Recall the basic AKLB setup: A is a Dedekind domain with fraction field K, L is a finite, separable extension of K of degree

### 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

### The Mathematics of Origami

The Mathematics of Origami Sheri Yin June 3, 2009 1 Contents 1 Introduction 3 2 Some Basics in Abstract Algebra 4 2.1 Groups................................. 4 2.2 Ring..................................

### Factoring Polynomials

Factoring Polynomials Sue Geller June 19, 2006 Factoring polynomials over the rational numbers, real numbers, and complex numbers has long been a standard topic of high school algebra. With the advent

### FINITE GROUP THEORY. FOR COMBINATORISTS Volume one. Jin Ho Kwak. Department of Mathematics POSTECH Pohang, 790 784 Korea jinkwak@postech.ac.

FINITE GROUP THEORY FOR COMBINATORISTS Volume one Jin Ho Kwak Department of Mathematics POSTECH Pohang, 790 784 Korea jinkwak@postech.ac.kr Ming Yao Xu Department of Mathematics Peking University Beijing

### 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