Mathematics - Automorphisms Conjugate Elements Detailed Notes with Solved Exercises, Study notes of Mathematics

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Mathematics - "Automorphisms Conjugate Elements" Detailed Notes with solved Exercises for BS Mathematics, MSc Mathematics, and Competitive Exams Preparation Automorphisms and Conjugate Elements Introduction We start by recalling that by an automorphism we mean an isomorphism of a group G to itself. Also under permutation groups we noticed that the set of all permutations (1—1 onto maps) forms a group. We show now that set of all automorphisms also forms a group, the two being closely related. We intend studying a few results pertaining to these groups. To begin with we take up few examples of automorphisms. Example 1: Let G be a group, then the identity map J: G > G, s.t., (x) = x is trivially an automorphism of G. In fact, it is sometimes called the trivial automorphism of G. Example 2: Let Z = group of integer under addition then f[:Z2 52, s+. I(n)=-n is an automorphism as f(n) = f(m) > -n =-m>n=m> fis 1-1. Again, since for any 7 € Z, f(-—n) = n we find fis onto. Now f(7 + my) =—-(n +m) =-n-—m=f(n) + fm) shows f is a homomorphism and hence an automorphism. Example 3: If G be an abelian group and f: G > G be such that f(x) = x! then as f(xy) = yt =ytxt=x' yt =f@) SO), fis a homomorphism. Again fe) =f) > xt =y" >x=y> fis 1-1. fis clearly onto and hence an automorphism. Example 4: If G be a non-abelian group, then the above defined map f: G > G s.t., f(x) = x" is not an automorphism. Since G is non-abelian, Jx,yé Gs.t., xy # yx