Matrix groups and regular semigroups
File(s)
Published version
Author(s)
Liebeck, Martin
Ye, Zhengkun
Type
Journal Article
Abstract
Let Tn denote the transformation semigroup consisting of all maps from [n]→[n], where [n]={1,...,n}. In 2009, Araújo, Mitchell and Schneider gave a classification
of subgroups G ≤ Sn such that ⟨G,a⟩ is a regular semigroup for all a ∈ Tn, showing that forn ≥ 10,theonlypossibilitiesforG are An and Sn. We investigate the analogous problem for linear groups over finite fields, raised by Araújo and Cameron: classify the subgroups of GLn(q) such that ⟨G, A⟩ is a regular semigroup for all matrices A in the matrix semigroup Mn(q). We obtain a representation theoretic criterion for such groups G, and exhibit a connection with the action of G on the exterior powers of the underlying space Fn q. As a consequence we prove the existence of several families of subgroups G of GLn(q) for arbitrary n with the property that ⟨G, A⟩ is regular for all A, in contrast with the previously mentioned result for Tn⋅
of subgroups G ≤ Sn such that ⟨G,a⟩ is a regular semigroup for all a ∈ Tn, showing that forn ≥ 10,theonlypossibilitiesforG are An and Sn. We investigate the analogous problem for linear groups over finite fields, raised by Araújo and Cameron: classify the subgroups of GLn(q) such that ⟨G, A⟩ is a regular semigroup for all matrices A in the matrix semigroup Mn(q). We obtain a representation theoretic criterion for such groups G, and exhibit a connection with the action of G on the exterior powers of the underlying space Fn q. As a consequence we prove the existence of several families of subgroups G of GLn(q) for arbitrary n with the property that ⟨G, A⟩ is regular for all A, in contrast with the previously mentioned result for Tn⋅
Date Issued
2025-10-01
Date Acceptance
2025-05-15
Citation
Semigroup Forum, 2025, 111 (11), pp.499-511
ISSN
0037-1912
Publisher
Springer
Start Page
499
End Page
511
Journal / Book Title
Semigroup Forum
Volume
111
Issue
11
Copyright Statement
© The Author(s) 2025 Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
License URL
Publication Status
Published
Date Publish Online
2025-06-16