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

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$p$-groups with maximal elementary abelian subgroups of rank $2$. / Glauberman, George; Mazza, Nadia.

In: Journal of Algebra, Vol. 323, No. 6, 15.03.2010, p. 1729-1737.

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Glauberman, George ; Mazza, Nadia. / $p$-groups with maximal elementary abelian subgroups of rank $2$. In: Journal of Algebra. 2010 ; Vol. 323, No. 6. pp. 1729-1737.

Bibtex

@article{e8e0c8a6e4074139bf8d4abddd32a20e,
title = "$p$-groups with maximal elementary abelian subgroups of rank $2$",
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.",
keywords = "Finite p-groups, Elementary abelian p-subgroups, Endotrivial modules, Class-breadth conjecture",
author = "George Glauberman and Nadia Mazza",
note = "The final, definitive version of this article has been published in the Journal, Journal of Algebra 323 (6), 2010, {\textcopyright} ELSEVIER.",
year = "2010",
month = mar,
day = "15",
doi = "10.1016/j.jalgebra.2009.10.015",
language = "English",
volume = "323",
pages = "1729--1737",
journal = "Journal of Algebra",
issn = "0021-8693",
publisher = "ELSEVIER ACADEMIC PRESS INC",
number = "6",

}

RIS

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 -