
Normal subgroups
Hi everyone.
Let $\displaystyle N$ be a normal subgroup of $\displaystyle G$ of finite index $\displaystyle n$. Given an element $\displaystyle t\in{G}$, let $\displaystyle h$ be the least positive integer such that $\displaystyle t^h\in{N}$. Prove that $\displaystyle hn$; also show that, if $\displaystyle t$ is if finite order $\displaystyle r$, then $\displaystyle hr$
Thanks.

Re: Normal subgroups
consider the mapping G→G/N given by g→gN.
then t→tN, and so t^{k}→t^{k}N.
if 0 < k < h, then t^{k}N ≠ N (since t^{k} is not in N).
but t^{k}N = (tN)^{k}, so tN has order h in G/N.
so h divides G/N = [G:N] = n.
since g→gN is a homomorphism, the order of gN divides the order of g, if the order of g is finite.
thus the order of tN divides the order of t, that is h divides r.