Purely noncommuting groups

Ben Blum-Smith, Fedor A. Bogomolov

Research output: Contribution to journalArticlepeer-review


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
Issue number4
StatePublished - Dec 1 2019


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

ASJC Scopus subject areas

  • General Mathematics


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