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

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<mark>Journal publication date</mark>15/03/2010
<mark>Journal</mark>Journal of Algebra
Issue number6
Number of pages9
Pages (from-to)1729-1737
<mark>Original language</mark>English


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.

Bibliographic note

The final, definitive version of this article has been published in the Journal, Journal of Algebra 323 (6), 2010, © ELSEVIER.