# Thread: Group of prime power order with unique subgroup

1. ## Group of prime power order with unique subgroup

If a group of order $\displaystyle p^m$, where p is an odd prime, contains a unique subgroup of order $\displaystyle p^s$, for some $\displaystyle 0 <s <m$,
then G must be cyclic.

How to prove this ?. Plz help.

2. ## Re: Group of prime power order with unique subgroup

Suppose it were NOT cyclic. Then there exist elements a and b such that a is not a power of b and b is not a power of a. Show that the set of powers of a form a subgroup and the set of powers of b form a different subgroup.

3. ## Re: Group of prime power order with unique subgroup

but here the condition is "for some 0 < s < m " not "for all 0 < s < m ".

4. ## Re: Group of prime power order with unique subgroup

Hi,
I've attached a solution. Initially, I was going to provide all the details, but then I realized I was in essence proving the theorem quoted first. So I just quoted the theorem with a reference.