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

Glauberman, George and Mazza, Nadia (2010) $p$-groups with maximal elementary abelian subgroups of rank $2$. Journal of Algebra, 323 (6). pp. 1729-1737. ISSN 0021-8693

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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 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 has more than one component.