$p$-groups with maximal elementary abelian subgroups of rank $2$.

George Glauberman, Nadia Mazza

Abstract:
Let $p$ be an odd prime number and $G$ a finite $p$-group. We prove
that if the rank of $G$ is greater than $p$, then $G$ has no maximal
elementary abelian subgroup of rank $2$. It follows that if $G$ has
rank greater than $p$, then the poset $\CE(G)$ of
elementary abelian subgroups of $G$ of rank at least $2$ is connected
and the torsion-free rank of the group of endotrivial $kG$-modules is
one, for any field $k$ of characteristic $p$. We also verify the
class-breadth conjecture for the $p$-groups $G$ whose poset $\CE(G)$ has
more than one component.