Covering semisimple groups by subgroups. (Q2901399)

A cover for a group is a collection of proper subgroups whose union is the whole group. A cover is called irredundant if no proper sub-collection is also a cover and it is called maximal if all its members are maximal subgroups. For an integer \(n>2\), a cover \(\mathcal C\) is called a \(\mathfrak C_n\)-cover whenever \(\mathcal C\) is an irredundant maximal core-free \(n\)-cover for \(G\).NEWLINENEWLINE Groups having a \(\mathfrak C_n\)-cover, \(n\in\{6,7\}\), have been classified by \textit{A. Abdollahi, M. J. Ataei, S. M. Jafarian Amiri} and \textit{A. Mohammadi Hassanabadi}, [Commun. Algebra 33, No. 9, 3225-3238 (2005; Zbl 1117.20016)], and \textit{A. Abdollahi} and \textit{S. M. Jafarian Amiri}, [J. Pure Appl. Algebra 209, No. 2, 291-300 (2007; Zbl 1117.20017)]. Groups having a \(\mathfrak C_n\)-cover, \(n\in\{3,4,5\}\), have been classified by \textit{G. Scorza}, [Bollettino U. M. I. 5, 216-218 (1926; JFM 52.0113.03)], \textit{D. Greco}, [Rend. Accad. Sci. Fis. Mat., IV. Ser., Napoli 23, 49-59 (1957; Zbl 0166.28304)], and \textit{R. A. Bryce, V. Fedri} and \textit{L. Serena}, [Bull. Aust. Math. Soc. 55, No. 3, 469-476 (1997; Zbl 0883.20014)].NEWLINENEWLINE The paper under review belongs to a series of works due to the author toward classification of groups \(G\) having a \(\mathfrak C_8\)-cover.NEWLINENEWLINE The main result of the paper under review is: Let \(G\) be a group having a \(\mathfrak C_8\)-cover. Suppose that \(\{M_i\mid 1\leq i\leq 8\}\) be a maximal irredundant \(8\)-cover for \(G\), with core-free intersection \(D\). Also for each \(i\in\{1,\dots,8\}\), assume that \(|G:M_i|=\alpha_i\) such that \(\alpha_1\leq\alpha_2\leq\cdots\leq\alpha_8\). It is proved that if \(\alpha_1\leq\alpha_2\leq 4\) and \(\alpha_3=6\), then \(G\) is not semisimple, where a group is called semisimple if it has no non-trivial normal Abelian subgroup.