Finite Abelian Groups

1. The problem statement, all variables and given/known data

Let G be a finite group with N , normal subgroup of G, and a, an element in G.

Prove that if (a) intersect N = (e), then o(An) = o(a).





2. Relevant equations







3. The attempt at a solution

(aN)^(o(a)) = a^(o(a)) * N = eN = N, but is the least power such that (aN)^m = N. Assume m must divide o(a).



(aN)^((o(a)) = (aN)^ (mq +r) where 0 <= r < m,

However, ((aN)^m)-q * a(N)^(o(a)) = (a(N)^r)= N = (a(N)^r).

r < m so r= 0 and mq= o(a).

I am not sure how to continue however, am I even going in the right direction?



Thanks!!