×

Semisimplity condition and covering groups by subgroups. (English) Zbl 1228.20018

A set \(\mathcal C\) of proper subgroups of a group \(G\) is called a cover for \(G\) if its set-theoretic union is equal to \(G\). If the size of \(\mathcal C\) is \(n\), we call \(\mathcal C\) an \(n\)-cover for the group \(G\). A cover \(\mathcal C\) for a group \(G\) is called irredundant if no proper subset of \(\mathcal C\) is a cover for \(G\). A cover \(\mathcal C\) for a group \(G\) is called maximal if all its members are maximal subgroups of \(G\). A cover \(\mathcal C\) for a group \(G\) is called core-free if \(D=\bigcap_{C\in\mathcal C}C\) is core-free, i.e. \(D_G=\bigcap_{g\in G}g^{-1}Dg\) is the trivial subgroup of \(G\). A cover \(\mathcal C\) for a group \(G\) is called a \(\mathfrak C_n\)-cover whenever \(\mathcal C\) is an irredundant maximal core-free \(n\)-cover for the group \(G\).
Let \(G\) be a finite semisimple group, i.e., \(G\) has no non-trivial normal Abelian subgroup. Suppose further that \(G\) has a \(\mathfrak C_8\)-cover \(\{M_i\mid 1\leq i\leq 8\}\). Assume that \(|G:M_i|=\alpha_i\) and assume without loss of generality that \(\alpha_1\leq\alpha_2\leq\cdots\leq\alpha_8\).
The main results of the paper under review are the following. Proposition 1.3. (a) If \(\alpha_1\leq\alpha_2\leq 4\), then \(\alpha_3\leq 6\). (b) If \(\alpha_1\leq\alpha_2\leq 4\) and \(\alpha_3=6\), then \(\alpha_i=6\) for \(3\leq i\leq 8\) and also \((M_i)_G\neq 1\) for \(1\leq i\leq 8\). Furthermore, \((M_i)_G\cap(M_j)_G\cap(M_k)_G=1\) for \(3\leq i<j<k\leq 8\).
Theorem 1.4. If \(\alpha_3\leq 6\), then every minimal normal subgroup of \(G\) is isomorphic to \(\text{Alt}_5\) or \(\text{Alt}_6\), where \(\text{Alt}_n\) denotes the alternating group of degree \(n\).

MSC:

20D60 Arithmetic and combinatorial problems involving abstract finite groups
20D30 Series and lattices of subgroups
20E28 Maximal subgroups
20D05 Finite simple groups and their classification
20D06 Simple groups: alternating groups and groups of Lie type

Software:

GAP
PDFBibTeX XMLCite
Full Text: Link