To introduce the concept of a mathematical structure called an algebraic group. To illustrate group concepts, we introduce cyclic and dihedral groups.


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1 1 Section 6. Introduction to the Algebraic Group Purpose of Section: To introduce the concept of a mathematical structure called an algebraic group. To illustrate group concepts, we introduce cyclic and dihedral groups. Introduction The theory of groups is an area of mathematics which is concerned with underlying relationships of things, and arguably the most powerful tool ever created for illuminating structure, both mathematical and physical. The word group was first used by the French genius Evariste Galois in 1830, who wrote his seminal paper on the unsolvability of the 5 th order polynomial equation, the night before he was killed in a stupid duel at the age of 0. Other early contributors to the development of group theory were Joseph Louis Lagrange, ( ), Niels Abel ( ), AugustinLouis Cauchy ( ), Arthur Cayley ( , Camille Jordan ( ), Ludwig Sylow ( ) and Marius Sophus Lie ( ). Now, merely more than a century later, group theory has resulted in an amazing unification of areas of mathematics, including algebra and geometry, long thought to be separate and unrelated. It is often said that whenever groups make an appearance in a subject, simplicity is created from chaos. Group theory has played (and is playing) a crucial role for both chemists and physicists to penetrate the deep underlying relationships in our amazing world. Binary Operation and Groups A binary operation on a set A is a rule, which assigns to each pair of elements of A a unique element of A. Thus, a binary operation is simply a function f : A A A. Two common binary operations familiar to the reader are +, i which assign the sum a + b R and product aib R to a pair ( a, b) R R of real numbers. We now give a formal definition of a group.
2 Definition: An algebraic group g G (or simply group) is a set of elements with a binary operation, say, satisfying the closure property as well as the following properties: a, b G a * b G Associative: is associative, that is, for every a, b, c G, we have ( a b) c = a ( b c). Identity: G has a unique identity 1 e. That is, for any element a G we have a * e = e* a = a Inverse: Every element a G has a unique inverse. That is, for a G 1 there exists an element a 1 1 G that satisfies a * a = a * a = e. We often denote a group G with operation " " G. by {, } Often it happens that a b = b a for all a, b G. When this happens the group is called a commutative (or Abelian) group. We often denote the group operation a b as ab, or maybe something more suggestive like if the group operation is addition or closely resembles addition. A group is called finite if it contains a finite number of elements and the number of elements in the group is called the order of the group. If the order of a group G is n, we denote this by writing G = n If the group is not of finite order we say it is of infinite order. In Plain English Associative: The associative property tells us when we combine three elements a, b, c G (keeping them in the same order), the result is unchanged regardless of which two elements are combined first. There are examples of algebraic groups where the operation is not associative, the cross product of vectors in vector analysis is an example of a nonassociative operation, as well as the difference between two numbers 3, but by far the majority of binary operations in mathematics are associative. 1 It is not necessary to state that the identity is unique since it can be proven there is only one identity. In a more lengthy treatment of groups, we would define the existence of an identity and then prove it is unique, but here we assume uniqueness to shorten the discussion. Again, it can be proven that the inverse is unique so it is not really necessary to assume uniqueness of an inverse in the definition. 3 ( a b) c a ( b c).
3 3 Identity: The identity element of a group depends on the binary relation and is the unique element e G that leaves every element a G unchanged when combined with e. In the group of the integers Z with the binary operation + (addition), the identity is 0 since a + 0 = 0 + a = a for every integer a. If the binary operation is (multiplication), then the identity e is 1 since a 1 = 1 a = a for all a in the group. 1 a Inverse: The inverse of an element a depends (of course) on the element a, but also on the identity e and is an element such that when 1 1 combined with a yields the identity; that is aa = a a = e. For example, the inverse of an integer a with group operation addition + is its negative a since a + a = a + a =. ( ) ( ) 0 Table 1: Properties of Binary Operations Operation Associative Commutative Identity Inverse on P( A) Yes Yes Yes No on P( A) Yes Yes Yes Yes gcd on N Yes Yes No No + on R Yes Yes Yes Yes on R No No No No on Q Yes Yes Yes Yes min on R Yes Yes No No Example 1: Group Test Which of the following define a group on the set of integers? a) { Z,+} : Integers with the operation of addition. b) { Z, ( m + n) / } : Integers with operation of averaging two integers. c) { Z, } Integers with operation of taking the difference of two integers. Solution a) { Z,+} : We leave it to the reader to show {,+} b) { Z, ( m + n) / } Z is a group. See Problem 5. : Taking the average of two integers, say and 3, is not an integer, hence the averaging operation is not closed in Z. Hence Z with the averaging operation is not a group. There is no need to check the other properties required of a group.
4 4 c) { Z, } The integers Z with the difference operation is not a group since subtraction is not associative, i.e. m ( n p) ( m n) p. However, it does have an identity, 0 since 0 m = m 0 = m. Also, every integer has an m m = 0, m m = 0. Nevertheless, failure of the inverse, itself, i.e. ( ) associative property says it is not a group. Abstraction Abstraction reveals connections between different areas of mathematics since the process of abstraction allows one to see essential ideas and see the forest and not just the trees. This broad viewpoint can result in making new discoveries in one area of mathematics as a result of knowledge in other areas. A disadvantage might be that highly abstract mathematics is more difficult to master and tends to isolate mathematics from the outside world. Cayley Table The binary operation of a group can be illustrated by means of a Cayley 4 table as drawn in Table, which shows the products gig j of elements g i and g j of a group. It is much like the addition or multiplication tables the reader studied as a child, except a Cayley table can record any binary operation. A Cayley table is an example of a latin square, meaning that every element of the group occurs once and exactly once in every row and column. We examine the Cayley table to learn about the inner workings of a group. g1 g1 = e g g 3 g j = e e g 1 g g j g g g g g 3 gg j g g g 1 g g g g 3 gig j i i i Cayley Table for a Group Figure Example Below we illustrate the only groups of order and 3. Show they are both commutative. Find the inverse of each element in the group. Show that each group is associative. Convince yourself that the only groups of these orders are the ones given. Keep in mind every row and column of the multiplication i 4 Arthur Cayley ( ) was an English mathematician
5 5 table includes every element of the group exactly once. We leave this fun for the reader. Order * e a e e a a a e Example 3: The set G { a, b, c, d} illustrated by the Cayley table Order 3 * e a b e e a b a a b e b b e a = and binary operation * define a group a) Is there an identity element? If so, find it. b) Find the inverse of each element. c) Is the binary operation commutative? a * b* c = a * b * c d) Is ( ) ( ) Solution a) The identity is a since ab = ba = b, ac = ca = c, ad = da = d b) a = a, b = d, c = c, d = b c) yes, the multiplication table is symmetric around the main diagonal a * b* c = a * d = d and a * b * c = b* c = d. In general, there is d) yes, ( ) ( ) no quick way to verify the associative property like there is the commutative property. You have to check ALL possible arrangements to verify associativity. On the other hand, if one instance where associativity fails, then the binary operation * is not associative.
6 6 Example 4: : Klein 4Group G = e, a, b, c described by the multiplication table in Show that the set { } Figure 1 forms a group. This group is called the Klein 5 4group group, which is the symmetry group of a (non square) rectangle 6 studied in Section 6.1. There are exactly two distinct groups of order four, the Klein 4group and the cyclic group Z 4 which we will study shortly. e a b c e e a b c a a e c b b b c e a c c b a e Multiplication Table for the Klein fourgroup Figure 1 Proof: First observe all products of elements of G belong to G since the table consists only of elements of G. The hardest requirement to check is r s t = r s t, where r, s can be associativity, which requires we check ( ) ( ) 3 any of the elements e, a, b, c, which means we have 4 = 48 equations to check since each or the r, s, t in the associative formula can take on one of four values e, a, b, c. The computations can be simplified by observing the group is commutative (i.e. r s = s r for all r, s G ), which implies ( r r) r = r ( r r) so we have associativity when r = s = t. Other shortcuts tricks can be used (as well as computer algebra systems) to shorten the list of elements you must check. In this example, we observe that the group operation is simply the composition of functions and we can resort to the fact that composition of functions is associative. Finding the group identity e is easy since multiplying any member of the group by e, either on the left of right, does not change the member. You can see that the first row and column of the table are the same as the group elements themselves. 5 Felix Klein ( ) was a German geometer and one of the major mathematicians of the 19 th century. 6 Try interpreting the elements, e a of the Klein group are 0 and 180 degree rotations of a rectangle, and c, d the horizontal and vertical flips of the rectangle..
7 7 1 r Finally, to find the inverse of an element r simply follow along the 1 row labeled " r " until you get to the group identity e, then the inverse r is the column label above e. You could also do the same thing by going down the column labeled " r " until reaching e, then the row label at the left of e is 1 r. In the Klein fourgroup each element 1, a, b, c is its own inverse since the identity e lies along the diagonal of the multiplication table. Familiar Groups You are familiar with more groups that you probably realize. Table shows just a few algebraic groups you might have seen in earlier studies. Group Elements Operation Identity Inverse Z n Z addition 0 n + Q n m / n m, n > 0 Z k { 0,1,,..., n 1} R { 0} GL x nonzero real number a, b R R ( ) (, R) general linear group SL (, R) special linear group a b c d ad bc 0 a b c d ad bc = 1 multiplication 1 n / m multiplication mod n Abelian yes yes 0 n k yes multiplication 1 1/ x yes vector addition matrix multiplication matrix multiplication Common Groups Table ( 0,0 ) ( a, b) yes d b ad bc ad bc c a ad bc ad bc d c b a Notational Note: Repeated multiplication of an element g of a group by itself n result in powers of an element and are denoted by g, n = 1,,.... When n = 0 we define 0 g as the identity 0 g = e. Cyclic Groups (Modular ( Arithmetic) The most common and most simple of all groups are the cyclic groups, which are wellknown to every child who has learned to keep time. no no
8 8 Definition: A finite cyclic group ( Z ) n, of order n is a group that contains an element g Z called the generator of the group, such that n 3 n 1 {,,,,..., } g e g g g g = Z n. where powers of g are simply repeated multiplications 7,,... 3 g = g g g = g g of g ; that is For example, the three rotational symmetries {,, } e R R of an equilateral triangle form a cyclic group Z 3 with generator g = R10 (note that R = R, R = e ) Cyclic groups also describe modular (or clock) arithmetic, which is the type of arithmetic we carry out when keeping time on 4hour timepiece, where the numbers wrap around after 4 hours. However, unless you are in the military, your clock lists hours from 0 to 11 so telling time is done, as mod(1) = 3 A.M.. they say, modulo 1, as in 5 hours after 9 P.M. is ( ) This leads us to the cyclic group Z 1 with elements Z 1 = { 0,1,,3, 4,5, 6, 7,8,9,10,11} and the group operation on Z 1 exactly what you do when you keep time, that is a b = ( a + b)mod1 where mod 1 refers to computing a b by computing ( a + b) then taking its remainder after dividing by 1. (We denote the group operation by to remind us it is addition, only reduced modulo 1.) For example = mod1 which, related to a 1hour clock, translates into A.M is 5 ( ) hours after 9 P.M. The hours of the clock Z 1 and keeping time using this binary operation defines an Abelian group of order 1, called a cyclic group, whose Cayley table is shown in Table 3. 7 We use the word multiplication here, but keep in mind the group operation can mean any binary operation, even addition.
9 Cayley table for the cyclic group of 1 elements Table 3 Figure 3 shows various clocks that give rise to different cyclic groups. The cyclic group Z 6 consists of elements Z6 = { 0, r1, r, r3, r4, r5 } were as always e is the identity map and r j is rotation of the clock by 60 j degrees, j = 1,...,5. This set, along with the binary operation of doing one operation after another (function composition) forms a group with the following Cayley table. In other to make the table read faster, we have replaced the angles of rotation r j by the time on the hour hand, 1,,3,, 11.
10 10 Z Cyclic Group Of Order Z 3 Cyclic Group of Order 3 Z 4 Cyclic Group of Order 4 Cyclic Groups Z n Z 6 Cyclic Group of Order Z 1 Cyclic of Order 1 See Table 1 Cyclic Groups Figure 3
11 11 Example 5 (Symmetries of a Square) Figure 4 shows the eight symmetries of a square, called the octic group. and ar 70. a) Is the octic group commutative? Hint: Compare products R70a b) There are several subsets of the eight symmetries that form a group in their own right. These are called subgroups of the octic group. Can you find all ten of them? Solution a) The reader can check but R70a ar70. Hence, the octic group is not commutative. b) The 10 subgroups of the octic group are { } { } { } { } { } { } { } { } e, e, v, e, h, e, d, e, a, e, R, e, R, v, h, e, R. d, a, D The reader can visualize these symmetries and make Cayley tables for them. (See Problem 16.) These subgroups form a partially ordered set and can be put in the Hasse diagram shown in Figure 5.
12 1 Motion Symbol First and Final Positions No motion e Rotate 90 Counterclockwise R 90 Rotate 180 Counterclockwise R 180 Rotate 70 Counterclockwise R 70 Horizontal flip h Vertical flip v Antidiagonal flip a Diagonal flip d Symmetries of a Square Figure 4
13 13 Hasse Diagram for the Subgroups of the Octic Group D 4 Figure 5 Symmetry Groups of n gons: Dihedral Groups In Section 6.1 we saw how symmetries of a figure in the plane, which is a rigid motion which leaves the figure unchanged, can create new symmetries by following one symmetry after another. In this way, one creates an arithmetic of symmetries, where the composition of symmetries plays the role of multiplication, and the system contains identity elements, inverses and all the goodies on an arithmetic system. In other words, a group of symmetries, called the symmetry group of the figure. Every figure no matter how non symmetric has at least one symmetric group, namely the group consisting only of the identity or do nothing symmetry. The more symmetric a figure the more elements in its symmetry group. In Section 6.1 we saw that the (non square) rectangle had four symmetries; namely rotations
14 14 of 0 and 180 degrees, and a horizontal and vertical flip about the midlines, which constitute the Klein 4group. On the other hand, the more symmetric square has 8 symmetries in its symmetry group. (Can you find them?) Some figures have both rotational and flip symmetries. A polygon is called regular if all its sides have the same length and all its angles are equal. An equilateral triangle is a regular 3gon, a square is a regular 4gon, a pentagon is a regular 5gon and so on. The symmetry group of a regular n gon, which has n rotational and n flip symmetries, for a total of n symmetries, is called the dihedral group of the n gon and denoted by D n Can you find the 10 symmetries of the dihedral group D 5 of the pentagon drawn in Figure 5? Find the Symmetric Group D 5 Figure 5 We are getting ahead of ourselves, but in addition to the complete dihedral group of 10 symmetries of a pentagon, there is also a smaller group of five rotation symmetries, called a subgroup, which is a group in its own right, of the larger group of 10 symmetries as well as a subgroup of the 5 flip symmetries. Figure 6 shows commercial figures whose symmetry groups are the dihedral groups D1 D5. symmetries D 1 one rotation (0 degrees), one vertical flip 4 symmetries D two rotations, two flips (horizontal and vertical axes)
15 15 D 3 three rotations, three flips 6 symmetries D 4 four rotations, four flips 8 symmetries D 5 five rotations, five flips 10 symmetries Figure with Dihedral Symmetry Groups Figure 6
16 16 Problems 1. Do the following sets with given binary operations form a group? If it does, give the identity element and the inverse of each element. If it does not form a group, say why. a) All even numbers, addition 1,1, multiplication b) { } c) All positive real numbers, multiplication d) All nonzero real numbers, division e) All real matrices, matrix addition f) the four numbers 1, 1, i, i where i = 1 is the unit complex number, the binary operation is ordinary multiplication. g) { a, b, c } with operation defined by the Cayley table a b c a b c a b a b c c c a b. (Finish the Group) Complete the following Cayley table for a group of order three. e a b e e a b a a b b 3. (Finish the Group) Complete the following Cayley table for a group of order four without looking at the Cayley tables of the Klein 4 group or the cyclic group of order 4 in the text. e a b c e e a b c a a b b c c 4. (Verification of a Group) Do the nonzero integers with the operation of multiplication form a group?
17 17 (Group You are Well Familiar) Show that {,+} 5. (Group You are Well Familiar) integers with the operation of addition) is a group. Z is a group (i.e. the 6. (Property of a Group) Verify that for all elements a, b in a group the identity ( ) ab = b a holds. Hint: Show that ( ab)( a b ) = e. General Properties of Groups 7. Show that a group has exactly one identity. 8. Show that every element in a group has no more than one inverse. 9. Show that in a group the identity ( ) (( ab c) d ) ( ab)( cd ) = holds. 10. Let a, b, c are members of a set with a binary operation on the set. If a = b, then the multiplication rule says ca = cb. a) Show that in a group we can cancel the c. b) Given an example of elements in the set Z 6 = { 0,1,,3, 4,5} with addition modulo 6, where cancellation does not hold. 11. (Cyclic Group) The cyclic group of order 6 describes the rotational symmetries of a regular hexagon. Its Cayley table is shown in Table 4. For notational simplicity the group elements are the number of degrees required to map the hexagon back onto itself Cayley Table for Z 6 Table 4 a) What is the inverse of each element of the group? b) The order of an element of a group is defined as the (smallest) number of repeated operations on itself that results in the group identity. What is the order of each element of the group?
18 18 c) Show that by taking repeated operations with itself of the element 40 the set of elements obtained is itself a group. Construct the Cayley table of this group. 1. (Affine Group) The set G of all transformations from the plane to the p lane of the form x = ax + by + c y = dx + ey + f where a, b, c, d, e, and f are real numbers satisfying ad bc 0, is a group if we define the group operation of performing one operation after the other, this forms a group called the affine group. a) What is the identity of the affine group? That is, what are the values of,,,,, a b c d e f? b) What is the inverse of( 0,0 )? 13.. (Mod 5 Multiplication) Create the multiplication table for the integers mod 5 arithmetic. Show this defines 0,1,, 3, 4 where multiplication defined as ( ) a group. 14. (Mod 4 Multiplication) Create the multiplication table for the integers mod 4 and show that this does not define a 0,1,,3 for modular arithmetic ( ) group. In other that the numbers 0,1,,..., n 1 forms a group under mod ( n) multiplication, it must be true that n is a prime number. 15. (Subgroups of the Dihedral Group D 4 ) A subgroup of a group is a subset of the elements of a group which itself a group using the group operation of the larger group. The Hasse diagram for the subgroups of the symmetries of a square (i.e. the dihedral group ) is shown in Figure 6. The letter " F " represents the identity element, the other letters represent rotations and reflections of F. Interpret the rotations in the subgroups and make a Cayley table for them. 16. (Subgroups of the Octic Group) Make a Cayley table for the subgroups of the octic group. a) { e, v } b) { e, h } c) { e, d }
19 19 d) { e, a } e) { e, R 180} f) { e, R180, v, h } g) { e, R, d, a } 180 Hasse diagram for the Subgroups of D 4 Figure (Isomorphic Groups) Sometime two groups appear different but are really the same group. The two groups in Figures 7, called Group A and Group B look different, but are really the same, or what are called isomorphic groups. Convince yourself the two groups are the same by making the substitution, or isomorphism that sends Group A into Group B., 0 i, 1 a, b, 3 c
20 Group A i a b c i i a b c a a b c i b b c i a c c i a b Group B 18. (Harder to See Isomorphic Groups) Show that Group C and Group D are isomorphic (the same group) by making the substitution, 0 1, 1, 4, 3 3 in Group C, and then interchanging the 3 rd and 4 th columns, followed by the 3 rd and 4 th rows of the resulting table Group C Group D
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