# Group Actions and Transitive Permutations

A) Show that if n is odd then the set of all n-cycles consists of two conjugacy classes of equal size in An

b) Let G be a transitive permutation group on the finite set A with |A|>1. Show that there is some g in G such that g(a) is not equal to a for all a in A. (Such an element g is called a fixed point free automorphism)

c)Let G be a group, let A be an abelian normal subgroup of G, and write

G(bar)=G/A. Show that G acts(on the left) by conjugation on A by

g(bar)a=gag^-1, where g is any representative of the coset g(bar). Give an explicit exxample to show that this action is not well defined if A is not abelian.

Ps. For part c G(bar) stands for notation of G with a bar(line) on the top....I don't know how to better type this notation on this screen.

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#### Solution Summary

Group Actions and Transitive Permutations are investigated.