Purely noncommuting groups

Ben Blum-Smith, Fedor A. Bogomolov

Research output: Contribution to journalArticle

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.

Original languageEnglish (US)
Pages (from-to)1173-1191
Number of pages19
JournalEuropean Journal of Mathematics
Volume5
Issue number4
DOIs
StatePublished - Dec 1 2019

Fingerprint

Supersolvable Group
Metabelian group
Commutator subgroup
Finite Simple Group
Projective Variety
Quotient
Permutation
Subgroup
Singularity
Distinct
Class

Keywords

  • Finite simple group
  • Linear representation
  • Metabelian group
  • Noncommuting operators
  • Shared eigenvector
  • Supersolvable group

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Purely noncommuting groups. / Blum-Smith, Ben; Bogomolov, Fedor A.

In: European Journal of Mathematics, Vol. 5, No. 4, 01.12.2019, p. 1173-1191.

Research output: Contribution to journalArticle

Blum-Smith, Ben ; Bogomolov, Fedor A. / Purely noncommuting groups. In: European Journal of Mathematics. 2019 ; Vol. 5, No. 4. pp. 1173-1191.
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