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Introduction to Group Theory (cont.) 1. Product Notation The product of n elements a1, a2, … an of a set G with multiplication is inductively defined as follows: 1 a i 1 1 a1 if n 1; n a i 1 i n 1 ( ai )an if n 1. i 1 This product is also written as a1·a2· … ·an. We have the following generalization of the associative law. Proposition 1: Let n0, n1, … nr be integers such that 0 = n0 < n1 < … < nr = n. Then nj n a ai . k j 1 k n j 1 1 i 1 r This is clear for r = 1. Hence we can assume that it is true for r 1 and prove it for r factors. The details are left as a homework exercise. 2. Groups Definition: A set G with a multiplication is called a group if the following (group) axioms are satisfied: [G1] The set G is not empty. [G2] If a, b, c G, then (ab)c = a (bc). [G3] There exists in G an element e such that (1) For any element a in G, ea = a. (2) For any element a in G there exists an element a’ in G such that a’a = e. In view of axiom [G2] and the general associativity law, we can and will write the product of any finite member of elements of G without inserting parentheses. Proposition: If G is a group and e the element specified in [G3(1)], then e is an identity element. Proposition: If G is a group, a an element of G, then a’ (specified in [G3(2)] is its inverse element. Proposition: If G is a group, then each of the equation ax = b and xa = b in an unknown x has a unique solution. 3. Examples for Groups The following are groups: {0,1} with multiplication given by the following table: · 0 1 0 0 1 1 1 0 {0,1,2} with multiplication given by the following table: · 0 1 2 0 0 1 2 1 1 2 0 2 2 0 1 {0,1,2,3} with multiplication given by the following table · 0 1 2 3 0 0 1 2 3 1 1 0 3 2 2 2 3 0 1 3 3 2 1 0 {0,1,2,3} with multiplication given by the following table · 0 1 2 3 0 0 1 2 3 1 1 2 3 0 2 2 3 0 1 3 3 0 1 2 4. Subgroups Definition: Let (G, ·) be a group and (H,) another one. Then (H,) is a subgroup of (G, ·) if and only if H G and a,b H: ab = a·b. With other words, a subgroup of G is a subset with the same multiplication that is itself a group. Proposition: Let H G be a subgroup. Then The identity element of H is the identity element of G . Proposition: A non-empty subset H of G is a subgroup (with the same multiplication) iff a, b H : ab 1 H.