\(p\)-groups with maximal elementary Abelian subgroups of rank 2.
Let \(G\) be a finite \(p\)-group with the prime \(p\) odd. The main result of the paper is that if the group \(G\) possesses a maximal elementary Abelian subgroup of rank 2 then the group \(G\) has rank at most \(p\), answering a question raised by the second author [in J. Algebra 320, No. 12, 4242-4248 (2008; Zbl 1165.20016)]. For the 2-group case it is shown that the group, with a maximal elementary Abelian subgroup of rank 2, should have rank at most 4, and an example is given that indeed there is such a 2-group of rank 4. Considering the poset of elementary Abelian subgroups ordered by inclusion, the main result has the consequence that if a finite group \(H\) has \(p\)-rank greater than \(p\) then the group of endotrivial \(kH\)-modules with \(k\) a field of characteristic \(p\) has torsion-free rank 1, applying the classification of endotrivial modules settled by \textit{J. F. Carlson} and \textit{J. Thévenaz} [Invent. Math. 158, No. 2, 389-411 (2004; Zbl 1110.20004)]. In connection with the class-breadth conjecture, recently studied by \textit{B. Eick, M. F. Newman, E. A. O'Brien}, [J. Algebra 300, No. 1, 384-393 (2006; Zbl 1108.20014)], the main result has the consequence that if for a finite \(p\)-group the poset of elementary Abelian subgroups has more than one component then the class-breadth conjecture is satisfied, that is, the group is of class not exceeding the class-breadth by more than one.
scientific article; zbMATH DE number 5687634
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