1.1 REVISION Definition 1.1.1: A group is a pair G, consisting of a non-emptey set G and a binary operation defined on G ,satisfying the following four properties: 1. G is closed under the operation . 2. The operation is associative. 3. G contains an identity element e for . 4. Each a ∈ G has an inverse a−1 ∈ G. Remark 1.1.2: If it happens that the group operation satisfies the commutativity property, then G, is called a commutative group or an abelian group. Example 1.1.3: Let K4 e, a , b , c .Define on K4 by e a a e a , e b b e b e c c e c , c b b c a a b b a c , a2 b2 c2 e Construct the group table? Solution : eabc eeabc aaecb bbcea ccbae Thus K4 is an abelian group which is called Klien -4-group. Definition 1.1.4: 1 Let H be a non-empty subset of the group G. Then H is said to be a subgroup of G if H itself a group under the same operation as that of G,we denote it by H ≤ G. Theorem 1.1.5: Let G be a group and ∅≠H ⊆ G. Then H ≤ G a,b ∈ H ab−1 ∈ G. Definition 1.1.6: The center of a group G , denoted by ZG, is the set ZG a ∈ G : xa ax ∀ x ∈ G. Theorem 1.1.7: Let G be a group. Then ZG ≤ G. Theorem 1.1.8: let H be a non-empty subset of a group G. Then H ≤ G i H is closed ab ∈ H ∀ a,b ∈ H ii a−1 ∈ H for each a ∈ H. _____________________________________________________________ Exercise 1.1.9: 1. Let G be a group and a ∈ G .DefineCa x ∈ G : xax−1 a.Show that Ca ≤ G which is called the centralizer of an element a. 2. Assume that H, K are subgroups of the commutative group G.Let HK g ∈ G : g hk h ∈ H, k ∈ K.Prove that HK ≤ G. _____________________________________________________________ Definition 1.1.10: Let n be a fixed positive integer. Two integers a and b are said to be congruent modulo n, written a ≡ b mod n a − b is divisible by n. 2 Definition 1.1.11: For an arbitrary integer a,let a a x ∈ Z : x ≡ a mod n x ∈ Z : x a qn q ∈ Z We call a a the congruenss class modulo n, determined by a and refer to a as a representative of this class. Theorem 1.1.12: (1) For each ve integer n, the mathematical system Zn,⊕ forms a commutative group, known as the group of integers modulo n. (2) Zn,⊙ is a commutative semi-group, with identity 1 ∈ Zn. Definition 1.1.13: Let G be any non-empty set. A permutation on a set G is a bijection from G to itself. Denote SG the set of all permutations on G.Take G N 1,2,3,...,n. Then the set of all permutations on N will be denoted by Sn. Theorem 1.1.14: 1. The set SG is a group under function composition ∘. This group is called symmetric group on the set G. 2. The system Sn,∘ forms a group known as the symmetric group on n symbols,whichis non-commutative for n ≥ 3. 3. An,∘ form a group of order n!/2 which is called the alternating group of n-letters, where An ∈ Sn : is an even permutation. Definition 1.1.15: A group G is called a cyclic group if there exist at least one element a ∈ G such that G an : n ∈ Z , if the operation is . G na : n ∈ Z , if the operation is . The element a is called the generator of G. The cyclic group generated by a is denoted by 〈a. 3 Example 1.1.16: (Infinite cyclic) Consider the group Z,.Then Z, is a cyclic group generated by 1, − 1. Example 1.1.17: (Finite cyclic) The additive group Zn,⊕ is a cyclic group with generator 1 . Definition 1.1.18: (i) Let G be a group . The order of a ∈ G is the smallest ve integer n such that an e na e if the operation is () .We denote it by oa n 0. (ii) If there is no such integer we say that a has infinite order denoted by oa . Definition 1.1.19: If G is a group such that for a ∈ G,G am : m ∈ Z,then G is a cyclic group. Denote it by G a . Theorem 1.1.20: Let G be a cyclic group. Then (i) G is commutative (every cyclic group is commutative) (ii) If H ≤ G H is cyclic. (every subgroup of a cyclic group is cyclic). Theorem 1.1.21: Let a be an element in a group G .Ifn is the least ve integer such that an e, i.e., oa n. Then (i) a has order n,and a e, a, a2, ..., an−1. (ii) If am e n ∣ m. (iii) If ak as k ≡ s mod n. Definition 1.1.22: The order of an element a oa of the group G is the order of the subgroup generated by a. That is oa o a . Theorem 1.1.23: Let G be a finite cyclic group of order n i.e., G a , |G| n with generator a ∈ G. Then (1) If d m,n is the greatest common divisor for the integer m,n, then the subgroup generated by am is the same as the subgroup generated by ad. 4 i.e., ad am da ma and am contains n ∣ d elements. (2) The distinct subgroups of G are those subgroups ad da if the operation is (), where d is a ve divisor of n. (3) If 1 m,n i.e., m,n are relatively prime, then am is a generator of G. am if the operation is . i.e., G . ma if the operation is () 5 Theorem 1.1.24: Let G a be an infinte cyclic group. i.e., oa . Then 1. If n ≠ m ∈ Z an ≠ am. 2. a, a−1 are the only generators of G. Definition 1.1.25: Let H ≤ G and a ∈ G. The left coset of H in G is defined as follows: aH ah : h ∈ H if the operation is . a H a h : h ∈ H if the operation is . Theorem 1.1.26: Let H ≤ G. Then there is a one-one correspondence between H and aH, for a ∈ G. i.e., |H| |aH|. Definition 1.1.27: Let H ≤ G, a,b ∈ G. Then a and b are left (or right )congruent modulo H denote it by a ≡l b ≡r −1 −1 a ≡l b mod H ab ∈ H a ≡r b mod H ba ∈ H. Theorem 1.1.28: Let H ≤ G. (i) Then the relation ≡l mod H is an equivalence relation on G. (ii) The equivalence classes are left cosets of H in G. a b ∈ G : a ≡l b mod H b ∈ G : a−1b ∈ H b ∈ G : b ∈ aH aH. Remark 1.1.29: (1) The set of left cosets of H in G partition G. i.e., G is the union of all left cosets of H in G and distinct left coets are disjoint. 6 (2) If b ∈ a a b. i.e., if b ∈ aH aH bH. (3) Let H ≤ G . Then aH bH a−1b ∈ H. Definition 1.1.30: Let H ≤ G. Then the index of H in G is defined to be the number of distinct left cosets of H, which is denote it by G : H. Theorem 1.1.31 (Lagrange Theorem): Let G be a group of finite order n, i.e., |G| n, H be a subgroup of G of order k. Then every subgroup of G is a divisor of |G|. i.e., |H| ∣ |G| and |G| |H|G : H. Proof : Let |G| n and |H| k. Then by theorem 1.1.26, if a ∈ G |H| |aH| k. G is the union of all disjoint left cosets of H in G. i.e., G a1H a2H a3H ... arH ,ai ∈ G where aiH ∩ ajH ∅ for i ≠ j. i.e., G : H r |G| |a1H| |a2H| ... |arH| n kr k ∣ n |H| ∣ |G|. Corollary 1.1.32: (i) Let |G| n. Then the order of any element a ∈ G divides n,andan e. (ii) Any finite group of prime order has no-non-trivial subgroup. (iii) Every group G of prime order is cyclic. (iv) Any non-commutative group has at least six elements. Definition 1.1.33: A subgroup H of a group G is said to be normal in G if every left coset of H in G is also a right coset of H in G.i.e., H G aH Ha ∀ a ∈ G. Remark 1.1.34: 7 1. G , e are normal. (Trivial normal subgroups} 2. Every subgroup of a commutative group is normal. 3. Every subgroup of a cyclic group is normal. 4. aH Ha it does not require that ah ha ∀ h ∈ H. Theorem 1.1.44: The subgroup H is normal in G aHa−1 ⊆ H for each element a ∈ G. i.e., aH Ha aHa−1 ⊆ H ∀ a ∈ G. Theorem 1.1.45: Let G be a group. Then (1) If H ≤ G s.t. H ⊆ ZG H G. (2) ZG G. (3) Every subgroup of a commutative group is normal.