Assignment:
Qa) Show that if n is odd then the set of all n-cycles consists of two conjugacy classes of equal size in An
Qb) 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)
Qc)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.
Provide complete and step by step solution for the question and show calculations and use formulas.