# Group of prime power order with unique subgroup

• Apr 23rd 2013, 09:39 AM
thippli
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.
• Apr 23rd 2013, 10:37 AM
HallsofIvy
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.
• Apr 24th 2013, 02:17 AM
thippli
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 ".
• Apr 24th 2013, 11:58 AM
johng
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.

Attachment 28124

Attachment 28125