Chapter-1 GROUPS 1.1. INTRODUCTION The theory of groups arose from the theory of equations, during the nineteenth century. Originally, groups consisted only of transformations. The group of transformations was later extended, and generalised to the concept of an abstract group. The abstract group is defined by a set of axioms. Groups arise naturally in various mathematical situations. They have played a large role in finding new connections between various branches of mathematics. They have helped reduce the study of some problems in geometry to the study of the corresponding problems in algebra. They have also found applications in physical sciences and biological sciences. The structure of a group is one of the simplest mathematical structures. Hence, groups may be considered as the starting point of the study of various algebraic structures. In this chapter, we shall define groups by a set of axioms, and study some of their properties. 1.2. BINARY OPERATION The definition of a group, by a set of axioms, involves the concept of a set with binary operation. In this section, we shall define a set with binary operation, and study some general properties of sets with binary operations. Intuitively, a set with binary operation is a set in which an abstract product is defined such that the product of two elements of the set is again an element of the set. Definition 1.2.1. A non-empty set G, together with a mapping f : G G G is called a set with binary operation. G is called the underlying set, and the mapping f : G G G is called the binary operation on the set G. Usually, we use the symbol a * b to denote f(a, b). The usual algebraic operations in real numbers, such as addition, subtraction or multiplication, are examples of binary operations in the set of real numbers. Division is also a binary operation in the set of non-zero real numbers. One can think of binary operations which are different from these usual algebraic operations. EXAMPLES 1. Let G be the set of integers. The operation f : G G G defined by f(a, b) = a * b where a * b = a + b ab, is a binary operation in G. We note that the above example does not define a binary operation in the set of positive integers, as for any two positive integers a and b, a * b = a + b ab may not always be a positive integer. 1
2 UNIVERSITY ALGEBRA 2. Let G be the set of all subsets of real numbers. For A G, and B G, the operation A * B = A U B is a binary operation in G. Similarly, the operation A * B = A B, is also a binary operation in G. We know that the binary operations of addition and multiplication in the set Z of integers have the following properties. For integers a, b and c, a + (b + c) =(a + b) + c (associativity for addition) a + b = b + a (commutativity for addition) a(bc) =(ab) c (associativity for multiplication) ab = ba (commutativity for multiplication) However, any binary operation will not always have these properties. We shall now define commutative, and associative binary operations. Definition 1.2.2. A binary operation * in G is said to be associative, if a * (b * c) = (a * b) * c, for any a, b and c in G. Definition 1.2.3. A binary operation * in G is said to be commutative, if a * b = b * a, for any a G and b G. As we had already mentioned, the usual operations of addition or multiplication in the set of integers, is both associative and commutative. Similarly the operation of U or on the set of subsets of a set is both commutative and associative. If we consider the binary operation, a * b = a b in the set of integers, this operation is not associative because (a b) c a (b c). It is also not commutative, because a b b a, for a and b, a b. The operation defined in Example 1 is commutative, viz. a * b = a + b ab = b + a ba = b * a. It is also associative because (a * b) * c=(a + b ab) * c = a + b ab + c (a + b ab) c = a + b + c ab ac bc + (ab) c a * (b * c) =a* (b + c bc) = a + b + c bc a(b + c bc) = a + b + c bc ab ac + a(bc). Thus (a * b) * c=a* (b * c). The operations of addition and multiplication in the set of integers have a neutral element, in the following sense. The integer 0 has the property that a + 0 = a = 0 + a for each a (neutral element for addition) and also a1 = 1a = a (neutral element for multiplication). We now define a neutral element in a set with binary operation. Definition 1.2.4. Let G be a set with binary operation. An element e G is called, a neutral element or identity element, if a * e = a = e * a, for each a G. For instance, the empty set Φ is the neutral element for the operation U, and the whole set X is the neutral element for the operation U. The binary operation of subtraction has no neutral element because a 0 = a but 0 a = a. For the operation a * b = a + b ab, zero is the neutral element as a * 0 = a = 0 * a. While considering numbers, the operation of subtraction is usually considered as the inverse of addition, and for non-zero numbers, division is considered as the inverse of multiplication. This leads us to the following definition. Definition 1.2.5. Let G be a set with binary operation * and unit element e. An element a G is said to have an inverse with respect to * if there exists another element a G such that a * a = e = a * a.
GROUPS 3 For instance, in real numbers, the inverse of a with respect to addition is a. If we consider nonzero real numbers, the inverse a with respect to multiplication is 1/a. Now consider the set of non-zero real numbers, and define a new binary operation, viz., a * b = ½ab. This operation has the following properties: (i) It is associative because a * (b * c) = a * (½bc) = ¼a(bc) and (a * b) * c = (½ab) * c = ¼(ab)c. Thus a * (b * c) = (a * b) * c. (ii) There exists a unit element, viz., the element e = 2, as a * 2 = ½a2 = a and 2 * a = ½2a = a. (iii) Every a has an inverse a = 4/a, because a * a = ½aa = ½a 4/a = 2 = e and a * a = ½a a = ½4/aa = 2 = e, the unit element. We shall see later several examples of sets with binary operations satisfying all the properties listed above. EXERCISES 1. Let G = {1, 1, i, i} be the set of four elements. Are the following operations binary operations in G? (i) a * b = a + b (ii) a * b = a. b, where + and. are addition and multiplication in complex numbers. 2. Define a binary operation on the set of even integers which is different from addition, subtraction or multiplication. 3. How many binary operations can be defined on a set with two elements? 4. Define a binary operation on the set of positive integers by a * b = max {a, b}. Show that this operation is both associative and commutative. 5. Let G = {a, b, c, d} be a set with four elements. Define a binary operation in G by the following table. a b c d a a b c d b b c c d c c d d a d d b d c Fig. 1.1 where, for x G, y G, the product x * y is the element in the row containing x and column containing y. Investigate, whether the binary operation is (i) associative, (ii) commutative, (iii) has a unit element. 6. Define a binary operation on the set of non-negative integers by, m * n = m 2 + n 2. Does this operation have (i) a unit element? (ii) an inverse for every element? 7. Give examples of binary operations on a finite set which is (i) associative but not commutative, (ii) commutative but not associative. 8. Give an example of a set with two elements and an associative binary operation on it which is not commutative.
4 UNIVERSITY ALGEBRA 9. Show that, for the binary operations given below all the following properties are satisfied: (i) associativity, (ii) existence of unit element, (iii) existence of inverse for each element. (a) The set of all real valued continuous functions on [0, 1] for the operation f * g = h where h(t) =f(t) + g(t), 0 t 1. (b) The set of all ordered pairs of real numbers, for the operation, (a, b) * (c, d) = (ac bd, ad + bc). (c) The set G = {1, 1, i, i}, for the operation a * b = ab. 1.3. GROUPS In the last section, we saw various examples of sets with binary operation which are associative, which have a unit element, and in which every element has an inverse. These are only particular cases of the abstract group defined as follows. Definition 1.3.1. A group (G, *) is a non-empty set G, with a binary operation * satisfying the following axioms. (i) a*(b*c) = (a*b) * c, a, b, c G (associativity). (ii) There exists an element e G, such that a * e = a = e * a, a G (existence of unit element). (iii) For each a G, there exists an element a G, such that a * a = e = a * a (existence of inverse). EXAMPLES 1. The set R of real numbers, for the binary operation of addition, is a group, with 0 as identity element and ( a) as the inverse of a. The same is true of the set Z of integers or the set Q of all rational numbers or the set C of complex numbers. 2. The set R* of non-zero real numbers, for the binary operation of multiplication, is a group with 1 as identity element, and 1/a as the inverse of a. The same is true of the set Q* of non-zero rational numbers or the set C* of non-zero complex numbers. 3. G = {1, 1, i, i} is a group for multiplication. 4. Consider the following transformations of the plane into itself: (i) Reflection in the x-axis. (ii) Reflection in the y-axis. (iii) Rotation through 180 about the origin (counter-clockwise). Let G consists of transformations of the type (i), (ii) and (iii) or a combination of them, successively. Then, G is a group where the group operation is the operation of taking composite of transformations. The associativity is satisfied because the composition of transformations is associative. The identity transformation, which is the composite of (i) with itself, acts as the identity element. The inverse of each element of the type (i), (ii) or (iii), is again an element of the same type. Thus any composite of transformations of the type (i), (ii) or (iii) has an inverse. In the above example, the elements of the group are not numbers but transformations, i.e. the above group, is a group of transformations. Such transformation groups are useful in geometry. 5. Let Z m be the set of residue classes modulo m. Define a binary operation in Z m by, i + j = k where i + j k (mod m), with 0 k m 1. This binary operation in Z m is associative, as + is associative in Z. It has 0 as a unit element, because i + 0 i (mod m), and 0 + i i (mod m). Since i + (m i) 0 (mod m) and (m i) + i 0 (mod m), i + ( m i) = 0 = ( m i)
GROUPS 5 + i. Thus, Z m is a group. This group is called the group of residue classes modulo m. It is also denoted by the symbol Z m. From the definition of a group, we know that, a group always has a unit element, and an inverse for each element. We now show that these are unique. Suppose G has two unit elements e and e. Then, by putting a = e in the relation a * e = a = e * a, we have e * e = e = e * e. Similarly, by putting a = e in the relation a * e = a = e * a, we have e * e = e = e * e. Hence e = e * e = e. Similarly, suppose an element a has two inverses, a and a. Then a * a = e = a * a, and a * a = e = a * a. Now a = a * e = a * (a * a ) = (a * a) * a (associativity) = e * a = a. We shall use the notation a 1 to denote the unique inverse of a. We have seen examples of sets with binary operation which satisfy some, but not all the axioms for a group. The set Z of integers, for instance, with respect to the operation of multiplication is associative, and has a unit element 1, but not every element has an inverse. A set with binary operation which is associative, and which has a unit element is called a semigroup (with identity). Thus Z, for the operation of multiplication, is a semi-group with identity. Similarly the set of all mappings of any set S in itself is a semi-group, for the operation of taking composite. The identity transformation is the identity element. These are not groups. We shall now prove some elementary properties of groups which follow directly from the axioms. We shall use the notation ab to denote the composite of a and b (instead of a * b), Proposition 1.3.2. Let G be a group and a, b, c elements of G. Then ab = ac implies b = c (left cancellation law). Similarly ba = ca implies b = c (right cancellation law). Proof. Consider the relation ab = ac. Multiply by a 1, the inverse of a, on both sides. Then a 1 (ab) = a 1 (ac). By associativity, this becomes (a 1 a)b = (a 1 a)c, i.e. eb = ec. Hence, we have b = c. The right cancellation law can be proved similarly. Proposition 1.3.3. Let G be a group and a, b elements of G. Then the equations ax = b, and ya = b, have unique solutions in G. Proof. Consider the equation ax = b. Multiply by a 1 on both sides. Then a 1 (ax) = a 1 b. By associativity, we have (a 1 a)x = a 1 b, i.e. ex = a 1 b. Thus x = a 1 b is a solution of the equation ax = b. Now, we show that the solution is unique. Suppose x 1 and x 2 are two solutions. Then ax 1 = b = ax 2. By left cancellation law in G, we have x 1 = x 2. Similarly, we can show that the equation ya = b has a unique solution in G. Proposition 1.3.4. Let G be a set with binary operation which is associative. Assume that G has a right unit element, and every element of G has a right inverse. Then G is a group. Proof. Let e be the right unit element of G so that ae = a for every a G. Since a has a right inverse, there exists a G, such that, aa = e. In view of the existence of right inverse, G has right cancellation law. We shall show now that e is also a left unit element. For any a G, let a be the right inverse of a so that aa = e. Now (ea)a = e(aa ) = ee = e = aa. By the right cacellation of a, we have ea = a, for each a G. Thus e is the unit element of G. Now we shall show that a is also left inverse of a. Now ea = a = a e = a (aa ) = (a a)a. By the right cancellation of a, we have e = a a. Thus a is the inverse of a, showing that G is a group. Proposition 1.3.5. Let G be a set with binary operation which is associative. Assume that, for all elements a and b in G, the equations ax = b and ya = b have unique solutions in G. Then G is a group.
6 UNIVERSITY ALGEBRA Proof. We first show that G has a unit element. Consider the equation ax = a for some a G. By assumption, it has a unique solution x = e, i.e. ae = a. Now we shall show that for any b G, be = b. Let y be the unique solution of the equation ya = b. Now be = (ya)e = y(ae) = ya = b. Thus e acts as unit element on the right. Given a G, the equation ax = e has a unique solution a, i.e. aa = e. Thus, every element has a right inverse. This implies that G is a group by Proposition 1.3.4. Definition 1.3.6. A group G is called a finite group, if the underlying set G is a finite set. The number of elements in a finite group is called the order of the group and is denoted by the symbol o(g). A group which is not finite is called an infinite group. If the order of a finite group is not very high, the binary operation can be exhibited conveniently by a table. Thus, for the group defined in Example 3, Section 1.3, the binary operation is given by the table that follows. 1 1 i i 1 1 1 i i 1 1 1 i i i i i 1 1 i i i 1 1 Fig. 1.2 The tabular form is often useful for defining a finite group. 6. Let G = {e, a, b, c}. Define a binary operation in G by the table e a b c e e a b c a a e c b b b c e a c c b a e Fig. 1.3 Under this operation the product xy is the element in the row containing x and column containing y. For instance ab = c, a 2 = e, etc. All the group axioms are satisfied, with e as unit element and inverses of a, b and c are a, b and c respectively. This group is called the Klein 4-group. We know that R for addition, or R* for multiplication, satisfy in addition to the group axioms, the following commutativity property, viz., a + b = b + a (commutativity for addition), and ab = ba (commutativity for multiplication). However, we have not assumed the commutativity property in the axioms for a group, as many of the interesting groups do not have the commutativity property. Definition 1.3.7. A group G is called an abelian group, if ab = ba, for all a, b in G. Abelian groups are sometimes also called commutative groups.
GROUPS 7 All the groups given in Examples 1 to 5 of Section 1.3 are abelian groups. We now give some Examples of non-abelian groups. 7. Consider an equilateral triangle with vertices 1, 2 and 3 as shown in Fig. 1.4. Consider the following transformations of the triangle to itself. (i) The three rotations about the centre through, 0, 120 and 240 (counter-clockwise). (ii) The three reflections along the three bisectors. We call anyone of the above six transformations a symmetry. For the binary operation of composition of transformations, the product of two symmetries is again a symmetry. The symmetries form a group which has as its identity element, 1 the identity transformation (corresponding to 0 rotation). Rotations through 120 and through 240 are mutual inverses of each other. For each of the reflections, the inverse is itself. Thus, the symmetries form a group. It is a non-abelian group. The product of rotation through 120 with reflection along the bisector through the vertex 1 is the reflection along the bisector through 2, but the product in the reverse 0 order, i.e. the product of reflection along the bisector through 1, with rotation through 120, is the reflection along the bisector through 3. 8. The above example can also be interpreted as follows. Each 2 3 symmetry permutes the vertices of the triangle and each permutation of the vertices of the triangle defines a symmetry. The three rotations Fig. 1.4 correspond respectively, to the permutations 1 2 3 = 1 2 3, = 1 2 3 2 3 1 and = 1 2 3 3 1 2 and the three reflections along the bisectors through 1, 2 and 3 respectively, correspond to the permutations 1 2 3 = 1 3 2, = 1 2 3 3 2 1 and = 1 2 3 2 1 3. The product of the symmetries corresponds to the product of the permutations. Thus, the six permutations form a group. This group will be denoted by S 3. The multiplication table for S 3 is now given. Fig. 1.5
8 UNIVERSITY ALGEBRA 9. Now, we consider the symmetries of the square. Consider a square with vertices 1, 2, 3 and 4. 4 3 Consider the following transformations of the square in itself. (i) The four rotations about the centre through 0, 90, 180 and 270 0 respectively (counter-clockwise). (ii) The two reflections along the diagonals and the two reflections along 1 2 the horizontal and vertical bisectors. Any transformation of the above type is called a symmetry of square. The Fig. 1.6 product (composite) of two symmetries is a symmetry. The symmetries form a group, and the group axioms can be easily verified. Each symmetry determines a permutation of the vertices 1, 2, 3 and 4; the four permutations corresponding to the four rotations respectively are,, 1 2 3 4 2 3 4 1 and. The reflections along the diagonals 13 and 24 respectively 3 4 1 2 4 1 2 3 correspond to 1 2 3 4 1 4 3 2 and 1 2 3 4. The reflections along the horizontal and vertical 3 2 1 4 bisectors correspond respectively to and respectively. Thus, the above 4 3 2 1 2 1 4 3 eight permutations also form a group. This group is called the dihedral group D 4. We can write down the elements of D 4 explicitly as follows. Let e = 1 2 3 4 1 2 3 4 and a = 1 2 3 4 2 3 4 1. Then a2 = and a 3 = 3 4 1 2 4 1 2 3. If b = 1 4 3 2 ; then b2 = e. Thus, D 4 = {e, a, a 2, a 3, b, ba, ba 2, ba 3 }. It is easy to verify that ab = ba 1 ba. The group D 4 is not abelian. a b 10. Let G be the set of 2 2 matrices of the form, a, b, c and d being real. Then G is a d group for addition defined by a b d + a b d = a + a b + b + c with the zero matrix d + d 0 0 0 0 as identity element and the inverse of a b d being a b. Similarly, the set of all 3 3 c d matrices over reals is a group for addition. Both these groups are abelian. a b 11. Let G be the set of 2 2 matrices, a, b, c and d being real with non-zero determinant, d i.e. ad bc 0. Then G is a group for multiplication defined by
GROUPS 9 a b a b aa + bc ab + bd = d d a + dc cb + dd 1 0 a b with the identity matrix 0 1 as identity element and the inverse of d b d is Δ c a where 1 Δ =. This group is usually denoted by GL (2, R). It is a non-abelian group. ad bc Similarly, the set of all 3 3 matrices over reals with non-zero determinant is a group for multiplication. This group is denoted by GL (3, R). It is also non-abelian. 12. The set of all n n real matrices with non-zero determinant (non-singular matrices) is a group for multiplication. This group is denoted by GL (n, R). Similarly the set of all n n real matrices with determinant one is a group for multiplication denoted by SL (n, R). If we consider the set of all nonsingular n n matrices over reals whose inverse coincides with the transpose, it is a group called the orthogonal group 0(n, R). We shall now state and prove the laws of exponents in a group. Given three elements a, b, c in a group G, we can form their product in two ways, viz. a (bc) and (ab) c. By associativity, these two products are the same. Hence, we can write the product abc of three elements without ambiguity. Similarly, the product of any finite number of elements can be written without ambiguity. Definition 1.3.8. Let G be a group, n an integer, and a G. We define powers of a as follows. a n = aa... n times, if n > 0. a = e and a n = (a 1 ) m if n = m, m > 0. Proposition 1.3.9. The following laws of exponents are true in a group G. (i) a n a m = a n+m, a G, m, n integers. (ii) (a n ) m = a nm, a G, m, n integers. (iii) (ab) n = a n b n, a, b G, n integers, provided ab = ba. Proof. (i) Assume first that m > 0 and n > 0. Then, a n a m = (aa...n times) (aa...m times) = aa...(m + n) times = a n+m. If m < 0 and n < 0, we can repeat the argument by replacing a by a 1. If either m = 0 or n = 0 the result follows trivially, as a 0 = e. Finally, consider the case when n > 0 and m < 0. The proof is now by induction on n. Let m = k, k > 0. For n = 1, a.a m = a.(a 1 ) k = a.(a 1.a 1... k times) = a 1. a 1... (k 1) times = (a 1 ) (k 1) = a m 1. Assume now the result for (n 1), and all m < 0. We have a n a m = aa n 1 a m = aa n+m 1 (by induction) = a n+m. The case n < 0, and m > 0 can be treated similarly. Thus, we have a n a m = a n+m for all m, n Z. The relations (ii) and (iii) can be proved similarly. We leave this as an exercise. It is useful sometimes to write the group operation additively especially when the group is abelian. In such a case, 0 is used to represent the identity element, and the inverse of a is denoted by a 1. The element aa... n-times, which we had denoted by a n will now be denoted as na. With this notation, we have, for any abelian group G: (i) (m + n)a = ma + na, a G, m, n Z. (ii) m(na) = (mn)a, a G, m, n Z. (iii) n(a + b) = na + nb, a, b G, n Z. (The condition a + b = b + a is now automatically satisfied as G is an abelian group.)
10 UNIVERSITY ALGEBRA EXERCISES 1. If G is a group, a G and b G, show that (i) (a 1 ) 1 = a and (ii) (ab) 1 = b 1 a 1. 2. Suppose a group G has an element x such that ax = x for all a G. Show that G is the trivial group, containing only the identity element. 3. Let G be a group, a G and b G. Show that (aba 1 ) n = aba 1, if and only if b = b n. 4. An element a G is called idempotent if a 2 = a. Show that the only idempotent element of G is the unit element. 1 2 3 5. Find a solution of the equation ax = b in S 3, where a = 3 2 1 and b = 1 2 3 1 3 2. 6. Show that the set of all transformations of the type z az + b, ad bc 0, of the complex cz + d numbers in itself, is a group for the operation of composite of transformations. (This group is called the Möbius transformation group. It is useful in complex analysis) 7. Show that the set of six transformations of the type t t, t 1/t, t 1, t, t 1 1 t, t t 1 t and t form a group, for the operation of composite of transformations. (This group is the t t 1 group of cross ratios and it is useful in projective geometry.) 8. If G is a group such that a 2 = e for every a G, show that G is abelian. Is the same true if a 3 = e for all a G? 9. Show that G is an abelian group, if and only if (ab) 2 = a 2 b 2, for all a G, b G. 10. Let G be a finite group with even number of elements. Show that there exists at least one a in G, a e such that a 2 = e. 11. Let S be any set, and G(S) the set of all (1, 1) mappings of S onto S. Show that G(S) is a group, for the operation of composition of mappings. 12. Let D n = {e, a, a 2...a n 1, b, ba,...ba n 1 } be a set of 2n elements. Define the product in D n, by the relations a n = e, b 2 = e and ab = ba 1. Show that these relations define the product of any two elements in D n. Show further, that D n is a group. (D n is called the nth dihedral group.) For n = 3, it is the symmetric group S 3, and for n = 4, it is the group of symmetries of the square. 13. Interpret D n as a group of symmetries of a regular n-sided polygon. 14. Show that the set of all rotations in the 3-space given by (x, y, z) (x, y, z ) is a group for composition where x = a 1 x + b 1 y + c 1 z y = a 2 x + b 2 y + c 2 z z = a 3 x + b 3 y + c 3 z, with a2 i + b2 i + c2 i = 1, (1 i 3) a i a j + b i b j + c i c j = 0, i j