This is what I have so far:If N is a finite normal subgroup of a group G and if G/N contains an element of ordern, prove that G contains an element of ordern.

Let have ordernfor some . Then , so . Since N is finite, let . Then , so .

When I pursued this line of reasoning, I was trying to show that , and thus . However, I'm stuck here. Have I made a mistake somewhere, or am I missing something?

Thanks for any insights.