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Justabeginner
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Homework Statement
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).
Homework Equations
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!