Intersections of matrix algebras and permutation representation of \(\text{PSL}(n,q)\)
A question that has many applications in finite group theory is the following. Let \(G\) be a finite group that acts transitively on a \(G\)-set \(\Omega\). Let \(H\) be some special type of subgroup of \(G\). Under what conditions does \(H\) have a regular orbit on \(\Omega\)? There is also a version of this problem for module actions. Suppose that \(K\) is a finite field and \(M\) is a faithful \(KG\)-module. Under what conditions is the regular module \(KH\) a submodule of \(M\). This type of problem was already significant in the Hall-Higman paper of 1956, when \(K\) was a field of characteristic \(p\), \(G\) a \(p\)-solvable group, and \(H\) a cyclic \(p\)-subgroup. The authors of the paper under review address the regular orbit problem in the case that \(G\) is a subgroup of the projective general linear group \(\text{PGL}(n,q)\) containing the projective special linear group \(\text{PSL}(n,q)\), where \(q\) is an integral power of the prime \(p\) and \(H\) is a cyclic subgroup of \(G\). The \(G\)-set \(\Omega\) is assumed to be non-trivial, in the sense that \(\text{PSL}(n,q)\) does not act trivially on it. The main result of the authors is that \(H\) has a regular orbit on \(\Omega\) except in the three cases that one might have anticipated to be problematic, namely, when \(n=2\) and \(q=2\) or \(3\), or when \(n=4\), \(q=2\), in which cases there are exceptional actions corresponding to isomorphisms with the symmetric group \(S_3\) or the alternating groups \(A_4\) and \(A_8\). The authors also consider a module-theoretic version of their theorem. In this case, let \(G\) be a subgroup of the general linear group \(\text{GL}(n,q)\) containing the special linear group \(\text{SL}(n,q)\) and let \(H\) be an Abelian subgroup of \(G\) whose Sylow \(p\)-subgroup is cyclic. Let \(K\) be a field of characteristic \(0\) or a prime that does not divide \(|G|\). Let \(M\) be a non-trivial permutation module for \(KG\). Let \(H_0\) be the subgroup of \(H\) that acts trivially on \(M\). Then we may view \(M\) as an \(H/H_0\)-module and this contains a regular \(K(H/H_0)\)-submodule, unless we have the three exceptional values of \(n\) and \(q\) described in the previous paragraph. In the case that \(n=4\) and \(q=2\), exceptional behaviour occurs when \(|H|=15\) and \(\dim M=8\). As the title of the paper might suggest, the authors make use of ring-theoretic machinery, especially investigating subrings of certain matrix algebras. The paper makes good reading for its instructive use of linear algebra, field theory, permutation group theory and representation theory.
scientific article; zbMATH DE number 1441818
scientific article