Groups without faithful transitive permutation representations of small degree
The authors investigate the titular groups, i.e., study the obstacles to a group \(G\) having large core-free subgroups \(H\). (\(H\) is core-free in \(G\) if it contains no normal subgroup \(\neq 1\) of \(G\).) Of particular interest are Dedekind subgroups of a group \(G\); a subgroup \(D\) of \(G\) is Dedekind if every subgroup of \(D\) is normal in \(G\). The main result of the paper is Theorem 4.9. There is a constant \(c\) such that the following holds with \(f(k)=\exp(c(\log k)^6)\): Each finite group \(G\) has two Dedekind subgroups \(D_1\) and \(D_2\) such that any subgroup of \(G\) of order exceeding \(f(k(G))\) has nontrivial intersection with \(D_1\) or \(D_2\), where \(k(G)\) denotes the order of the largest core-free subgroup of \(G\). Related properties of \(p\)-groups and infinite locally finite groups without large core-free subgroups are discussed, and as an extension of a result due to Gaschütz and Itô (a finite group \(G\) with \(k(G)=1\) is solvable) the authors obtain Theorem 2.13: There is a constant \(c\) such that if group \(G\) has a solvable subgroup of index \(n\) then \(G\) has a solvable normal subgroup of index at most \(n^c\). The authors end by summarizing ten open questions/conjectures mentioned in the body of the paper. This feature alone should attract many readers!
scientific article; zbMATH DE number 1076264
scientific article