Section 1.5. Homomorphisms 39 Definition. Let a1,a2,...,an be a list with no repetitions of all the elements in a group G. A multiplication table for G is the n × n matrix whose ij entry is aiaj. G a1 a2 · · · aj · · · an a1 a1a1 a1a2 · · · a1aj · · · a1an a2 a2a1 a2a2 · · · a2aj · · · a2an ai aia1 aia2 · · · aiaj · · · aian an ana1 ana2 · · · anaj · · · anan A multiplication table for a group G of order n depends on the listing of the ele- ments of G, and so G has n! different multiplication tables. (Thus, the task of deter- mining whether a multiplication table for a group G is the same as a multiplication table for another group H is a daunting one, involving (n!)2 comparisons, each of which involves checking n2 entries.) If a1,a2,...,an is a list of all the elements of G with no repetitions, and if f : G → H is a bijection, then f(a1),f(a2),...,f(an) is a list of all the elements of H with no repetitions, and so this latter list determines a multiplication table for H. That f is an isomorphism says that if we superimpose the given multiplication table for G (determined by a1,a2,...,an) upon the multi- plication table for H [determined by f(a1),f(a2),...,f(an)], then the tables match: if aiaj is the ij entry in the multiplication table of G, then f(aiaj) = f(ai)f(aj) is the ij entry of the multiplication table for H. In this sense, isomorphic groups have the same multiplication table. Thus, isomorphic groups are essentially the same, differing only in the notation for the elements and the binary operations. Example 1.57. Let us show that G = S3, the symmetric group permuting {1, 2, 3}, and H = SY , the symmetric group permuting Y = {a, b, c}, are isomorphic. First, list G: (1), (1 2), (1 3), (2 3), (1 2 3), (1 3 2). We define the obvious function ϕ: S3 → SY that replaces numbers by letters: (1), (a b), (a c), (b c), (a b c), (a c b). Compare the multiplication table for S3 arising from this list of its elements with the multiplication table for SY arising from the corresponding list of its elements. The reader should write out the complete tables of each and superimpose one on the other to see that they do match. We will check only one entry. The 4, 5 position in the table for S3 is the product (2 3)(1 2 3) = (1 3), while the 4, 5 position in the table for SY is the product (b c)(a b c) = (a c). The same idea shows that S3 ∼ = D6, for symmetries of an equilateral triangle correspond to permutations of its vertices. This result is generalized in Exercise 1.46 on page 44. Lemma 1.58. Let f : G → H be a homomorphism of groups. (i) f(1) = 1. (ii) f(x−1) = f(x)−1. (iii) f(xn) = f(x)n for all n ∈ Z.

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