Generalized bases of finite groups

authored by

Benjamin Sambale

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 Δ of a finite group G is called a p-base (where p is a prime) if ⟨ Δ ⟩ is a p-group and C _G(Δ) is p-nilpotent. Building on results of Halasi–Maróti, 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.

Organisation(s)

Institute of Algebra, Number Theory and Discrete Mathematics

Type

Article

Journal

Archiv der Mathematik

Volume

117

Pages

9-18

No. of pages

10

ISSN

0003-889X

Publication date

07.2021

Publication status

Published

Peer reviewed

Yes

ASJC Scopus subject areas

General Mathematics

Electronic version(s)

https://doi.org/10.1007/s00013-021-01589-x (Access: Open)