 Abstract algebra

This section provides an overview of basic algebraic objects and their properties, for reference in the remainder of this handbook. Several of the definitions in §2.5.1 and §2.5.2 were presented earlier in §2.4.3 in the more concrete setting of the algebraic structure Z*.

2.161 Definition A binary' operation * on a set 5 is a mapping from S x S to S. That is, * is a rale which assigns to each ordered pair of elements from S an element of S.

Groups

• 2.162 Definition A group (G, *) consists of a set G with a binary operation * on G satisfying the following three axioms.
• (i) The group operation is associative. That is, a*(b*c) = (a*b)*c for all a, b. c e G.
• (ii) There is an element 1 € G, called the identity element, such that a * 1 = 1 * a = a for all a e G.
• (iii) For each a e G there exists an element a~lG, called the inverse of a, such that a * a-1 = a-1 * a = 1.

A group G is abelian (or commutative) if, furthermore,

(iv) a * b = b * a for all a, bG.

Note that multiplicative group notation has been used for the group operation. If the group operation is addition, then the group is said to be an additive group, the identity element is denoted by 0, and the inverse of a is denoted -a.

Henceforth, unless otherwise stated, the symbol * will be omitted and the group operation will simply be denoted by juxtaposition.

• 2.163 Definition A group G is finite if |G| is finite. The number of elements in a finite group is called its order.
• 2.164 Example The set of integers Z with the operation of addition forms a group. The identity

element is 0 and the diverse of an integer a is the integer —a.

2.165 Example The set Z„, with the operation of addition modulo n, forms a group of order n. The set Zn with the operation of multiplication modulo n is not a group, since not all elements have multiplicative inverses. However, the set Z* (see Definition 2.124) is a group of order o(n) under the operation of multiplication modulo n, with identity element 1. □

• 2.166 Definition A non-empty subset Я of a group G is a subgroup of G if II is itself a group with respect to the operation of G. If Я is a subgroup of G and II ф G, then Я is called a proper subgroup of G.
• 2.167 Definition A group G is cyclic if there is an element a e G such that for each b e G there is an integer i with b = a1. Such an element a is called a generator of G.
• 2.168 Fact If G is a group and a e G, then the set of all powers of a forms a cyclic subgroup of G, called the subgroup generated by a, and denoted by {a).
• 2.169 Definition Let G be a group and a e G. The order of a is defined to be the least positive integer f such that al = 1, provided that such an integer exists. If such a t does not exist, then the order of a is defined to be oo.
• 2.170 Fact Let G be a group, and let о € G be an element of finite order t. Then |{a)|, the size of the subgroup generated by a, is equal to t.
• 2.171 Fact (Lagrange’s theoreni)U G is a finite group and Я is a subgroup of G, thenH divides |G|. Hence, if a € G, the order of a divides |G|.
• 2.172 Fact Every subgroup of a cyclic group G is also cyclic. In fact, if G is a cyclic group of order n, then for each positive divisor d of n, G contains exactly one subgroup of order d,
• 2.173 Fact Let G be a group.
• (i) If the order of aG is t, then the order of ak is tj gcd(f, k).

(ii) If G is a cyclic group of order n and d , then G has exactly (d) elements of order d. In particular, G has ф(п) generators.

2.174 Example Consider the multiplicative group Z}9 = {1.2,... , 18} of order 18. The group

is cyclic (Fact 2.132(i)), and a generator is a = 2. The subgroups of Z}9, and their generators, are listed in Table 2.7. □

 Subgroup Generators Order {1} 1 1 {1,18} 18 2 {1,7,11} 7,11 3 {1,7,8,11,12,18} 8,12 6 {1,4,5,6,7,9,11,16,17} 4,5,6,9,16,17 9 {1,2,3,... ,18} 2,3,10,13,14,15 18

Table 2.7: The subgroups of Z{g.