Thread: Normal 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.

consider the mapping G→G/N given by g→gN.

then t→tN, and so t^k→t^kN.

if 0 < k < h, then t^kN ≠ N (since t^k is not in N).

but t^kN = (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.