Purely noncommuting groups

TL;DR

This paper introduces the class of purely noncommuting (PNC) groups, exploring their properties, examples, and implications for group actions on algebraic varieties, revealing connections between group structure and geometric actions.

Contribution

It defines the PNC property, characterizes which groups are PNC, and provides criteria and examples, including results on supersolvable, simple, and metabelian groups.

Findings

All supersolvable groups are PNC.

No nonabelian finite simple groups are PNC.

A metabelian group is PNC if its commutator factors are distinct prime powers.

Abstract

In this paper we define and investigate a class of groups characterized by a representation-theoretic property we call 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.