Hi everyone.

Let be a normal subgroup of of finite index . Given an element , let be the least positive integer such that . Prove that ; also show that, if is if finite order , then

Thanks.

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- Apr 10th 2012, 03:30 AMFernandoNormal subgroups
Hi everyone.

Let be a normal subgroup of of finite index . Given an element , let be the least positive integer such that . Prove that ; also show that, if is if finite order , then

Thanks. - Apr 10th 2012, 06:06 AMDevenoRe: 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.