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.
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.
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.