Order of a Group with a Unique Subgroup

Hello!

I have a problem I'm working on..

Given that a finite group G has exactly one nontrivial proper subgroup (call it H),

then we must prove

a) G must be cyclic

b) the order of G is for a prime p.

I have shown a) by noting that any element in G either generates

Case 1:the nontrivial proper subgroup H

Case 2:the group itself

So there is at least one element which generates the group (so it is cyclic).

I am having trouble showing part b), however.

Any help would be fantastic! and cause me to be very grateful! Thanks!

Cheers.