# Normal subgroups

• April 10th 2012, 02:30 AM
Fernando
Normal subgroups
Hi everyone.

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

Thanks.
• April 10th 2012, 05:06 AM
Deveno
Re: Normal subgroups
consider the mapping G→G/N given by g→gN.

then t→tN, and so tk→tkN.

if 0 < k < h, then tkN ≠ N (since tk is not in N).

but tkN = (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.