Generalized bases of finite groups

TL;DR

This paper introduces p-bases as a new local invariant for finite groups, proving their existence with small sizes for p-solvable groups and other groups, and conjecturing their universal small size.

Contribution

It defines p-bases for finite groups, proves their small size in p-solvable and certain groups, and extends the concept to blocks and fusion systems.

Findings

p-solvable groups have p-bases of size 3 for every prime p

Certain prominent groups have p-bases of size 2

Conjecture that all finite groups have p-bases of size 2

Abstract

Motivated by recent results on the minimal base of a permutation group, we introduce a new local invariant attached to arbitrary finite groups. More precisely, a subset Delta of a finite group G is called a p-base (where p is a prime) if Delta generates a p-group and C_G(Delta) is p-nilpotent. Building on results of Halasi-Mar\'oti, we prove that p-solvable groups possess p-bases of size 3 for every prime p. For other prominent groups we exhibit p-bases of size 2. In fact, we conjecture the existence of p-bases of size 2 for every finite group. Finally, the notion of p-bases is generalized to blocks and fusion systems.