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
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.