p-Groups with maximal elementary abelian subgroups of rank 2

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 epsilon(G) of elementary abelian Subgroups of C 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 epsilon(G) has inure than one component. (C) 2009 Elsevier Inc. All rights reserved.