Year: 2025
Type: article
Abstract: Let G be a group. The autocentre, or absolute centre, of G is the subgroup of G of all the elements fixed by every automorphism of G,while the autocommutator subgroup of G is the subgroup of G generated by all the elements g⁻¹g^(α), with g ∈ G and α ∈ Aut(G). In the present article we introduce some homological tools to study the properties of these subgroups and, by means of these, we extend several results for finite and infinite groups. In particular, we show that the torsion subgroup of the autocentre of G lies almost always inside the Frattini subgroup of G and give polynomial bounds for the order of the autocommutator subgroup of G, improving those previously given by Hegarty.
Source: Ars Mathematica Contemporanea
Author: Mattia Brescia
Institution: University of Naples Federico II
Language: English
Topic: Finite Group Theory Research
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: diamond
Awards: G1953887145, G6051190257
OpenAlex ID: W4414947808
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