Research output: Contribution to Journal/Magazine › Journal article › peer-review
Research output: Contribution to Journal/Magazine › Journal article › peer-review
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TY - JOUR
T1 - $p$-groups with maximal elementary abelian subgroups of rank $2$
AU - Glauberman, George
AU - Mazza, Nadia
N1 - The final, definitive version of this article has been published in the Journal, Journal of Algebra 323 (6), 2010, © ELSEVIER.
PY - 2010/3/15
Y1 - 2010/3/15
N2 - 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.
AB - 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.
KW - Finite p-groups
KW - Elementary abelian p-subgroups
KW - Endotrivial modules
KW - Class-breadth conjecture
U2 - 10.1016/j.jalgebra.2009.10.015
DO - 10.1016/j.jalgebra.2009.10.015
M3 - Journal article
VL - 323
SP - 1729
EP - 1737
JO - Journal of Algebra
JF - Journal of Algebra
SN - 0021-8693
IS - 6
ER -