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Purely noncommuting groups

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Abstract

We define and investigate a class of groups characterized by a representation-theoretic property, called purely noncommuting or PNC. This property guarantees that the group has an action on a smooth projective variety with mild quotient singularities. It has intrinsic group-theoretic interest as well. The main results are as follows. (i) All supersolvable groups are PNC. (ii) No nonabelian finite simple groups are PNC. (iii) A metabelian group is guaranteed to be PNC if its commutator subgroup’s cyclic prime-power-order factors are all distinct, but not in general. We also give a criterion guaranteeing a group is PNC if its nonabelian subgroups are all large, in a suitable sense, and investigate the PNC property for permutations.

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Notes

  1. Abelian groups themselves are PNC, vacuously. It is perhaps a failing of our nomenclature that abelian groups are called “purely noncommuting”.

  2. In fact, if \(\lambda = (\lambda _1,\dots ,\lambda _r)\) with \(\lambda _1\geqslant \cdots \geqslant \lambda _r\) is the partition describing the type of A, so that , then the tuple \((\dim A_0,\dots ,\dim A_{k-1})\) is the conjugate partition \(\lambda '\).

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Acknowledgements

The authors wish to thank Frieder Ladisch for the statement and proof of Lemma 6.8, and an anonymous referee for very helpful comments. With the exception of Sects. 7 and 8, this paper is a revision of work that originally appeared as part of the first author’s doctoral thesis under the supervision of the second author and Yuri Tschinkel at the Courant Institute of Mathematical Sciences at NYU.

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Authors and Affiliations

  1. Department of Natural Sciences and Mathematics, Eugene Lang College of Liberal Arts, 66 West 12th Street, New York, NY, 10011, USA

    Ben Blum-Smith

  2. Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY, 10012, USA

    Fedor A. Bogomolov

  3. AG Laboratory, National Research University Higher School of Economics, 7 Vavilova Street, Moscow, Russia, 117312

    Fedor A. Bogomolov

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  1. Ben Blum-Smith
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  2. Fedor A. Bogomolov
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Correspondence to Ben Blum-Smith.

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F.A. Bogomolov has been funded by the Russian Academic Excellence Project ‘5-100’ and acknowledges support by a Simons travel Grant and by the EPSRC program Grant EP/M024830.

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Blum-Smith, B., Bogomolov, F.A. Purely noncommuting groups. European Journal of Mathematics 5, 1173–1191 (2019). https://doi.org/10.1007/s40879-019-00348-1

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