Order of a Group with a Unique Subgroup
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!