Irredundant families of maximal subgroups of finite solvable groups

Document Type : Ischia Group Theory 2020/2021


Faculty of Mathematics, University of Bialystok, Ciolkowskiego 1M, 15-245 Bialystok, Poland


Let $\mathcal{M}$ be a family of maximal subgroups of a group $G.$ We say that $\mathcal{M}$ is irredundant if its intersection is not equal to the intersection of any proper subfamily of $\mathcal{M}$. The maximal dimension of $G$ is the maximal size of an irredundant family of maximal subgroups of $G$. In this paper we study a class of solvable groups, called $\mathcal{M}$-groups, in which the maximal dimension has properties analogous to that of the dimension of a vector space such as the span property, the extension property and the basis exchange property.


Main Subjects

[1] K. Archer, H. B. Serrano, K. Cook, L. K. Lauderdale, Y. Perez and V. Villalobos, On the intersection numbers of
finite groups, preprint, arXiv:1907.02898, 2019.
[2] C. Bagiński and A. Stocka, Finite groups with L-free lattices of subgroups, Illinois J. Math., 52 no. 3 (2008) 887–900.
[3] T. C. Burness, M. Garonzi and A. Lucchini, On the minimal dimension of a finite simple group, J. Combin. Theory
Ser. A, 171 (2020) 32 pp.
[4] T. C. Burness, M. Garonzi and A. Lucchini, Finite groups, minimal bases and the intersection numbers, preprint,
arXiv:2009.10137v1, 2020.
[5] E. Detomi and A. Lucchini, Maximal subgroups of finite soluble groups in general position, Ann. Mat. Pura Appl.
(4), 195 no. 4 (2016) 1177–1183.
[6] K. Doerk and T. Hawkes, Finite soluble groups, De Gruyter Expositions in Mathematics, 4, Walter de Gruyter &
Co., Berlin, 1992.
[7] R. Fernando, On an inequality of dimension-like invariants for finite groups, preprint, arXiv:1502.00360, 2015.
[8] M. Garonzi and A. Lucchini, Maximal irredundant families of minimal size in the alternating group, Arch. Math.
(Basel), 113 (2019) 119–126.
[9] P. Grzeszczuk and E. R. Puczyowski, On Goldie and dual Goldie dimensions, J. Pure Appl. Algebra, 31 (1984)
[10] J. Krempa and A. Sakowicz, On uniform dimension of finite groups, Colloq. Math., 89 (2001) 223–231.
[11] J. Krempa and B. Terlikowska-Oslowska, On uniform dimension of lattices, Contributions to general algebra, 9,
Hölder-Pichler-Tempsky, Vienna, 1995 219-230.
[12] D. J. S. Robinson, A course in the theory of groups, Second edition. Graduate Texts in Mathematics, 8, Springer-
Verlag, New York, 1996.
[13] R. Schmidt, Subgroup lattices of groups, De Gruyter Expositions in Mathematics, 14, Walter de Gruyter & Co.,
Berlin, 1994.
[14] M. Weidner, Independence and maximal subgroups, Illinois J. Math., 40 no. 1 (1996) 47–76.
[15] H. Whitney, On the abstract properties of linear dependence, Amer. J. Math., 57 no. 3 (1935) 509–533.
[16] T. Liu and R. Keith Dennis, MaxDim of some simple groups, preprint, arXiv:1712.04553, 2017.
[17] W. Niu, On Behaviors of Maximal Dimension, preprint, arXiv:1801.08327, 2018.
Volume 12, Issue 3 - Serial Number 3
Proceedings of the Ischia Group Theory (2020/2021) - Part 3
September 2023
Pages 163-176
  • Receive Date: 04 October 2021
  • Revise Date: 06 February 2022
  • Accept Date: 23 May 2022
  • Published Online: 01 September 2023