# A question about finitely generated $p$-groups

My question is about finitely generated $p$-groups. In general, a subgroup of a finitely generated group is not necessarily finitely generated. But, my question is about finite and finitely generated $p$-groups. More specifically: if $G$ is a finitely generated $p$-group, say, $m$-generated and $U$ is a finitely generated subgroup of $G$, then is $U$ at most $m$-generated? If not, can $U$ be generated by a number of elements that depends only on $m$?

• Is G finite? That's not 100% clear from the question. Mar 28, 2013 at 18:40
• I don't think this is true, take a maximal elementary abelian subgroup. Mar 28, 2013 at 18:42
• Yes, yatima2975. $G$ is finite.
– user59969
Mar 28, 2013 at 18:42

A wreath product of a cyclic group of order $p$ with a cyclic group of order $p^k$ is 2-generated, but the base group of the wreath product requires $p^k$ generators. So the answer to both questions is no.
• @AgenorAndrade DerekHolt's answer doesn't contradict Hall. If a group is $k$ generated it's also $m$ generated for $m> k$. Mar 28, 2013 at 20:46
• Proving that a group of order $p^k$ can be generated by at most $k$ elements is easy. Just keep choosing elements that are not in the subgroup generated by the elements chosen so far, and use Lagrange's Theorem. Mar 28, 2013 at 20:51