On 9- and 10-decomposable finite groups.
A finite group is called \(n\)-decomposable if every proper non-trivial normal subgroup of \(G\) is a union of \(n\) distinct conjugacy classes of \(G\). The first author with co-authors studied the question of finding all positive integers \(n\) such that there is an \(n\)-decomposable finite group. In this paper, the authors describe 9- and 10-decomposable non-perfect finite groups. They prove that a non-perfect finite group \(G\) is 9-decomposable if and only if \(G\) is isomorphic to \(\Aut(L_2(32))\), \(\Aut(L_3(3))\), the non-Abelian semi-direct product \((Z_5\times Z_5):Z_3\) or a non-Abelian group of order \(pq\), where \(p\) and \(q\) are primes and \(p-1=8q\), and also a non-perfect finite group \(G\) is 10-decomposable if and only if \(G\) is isomorphic to \(\Aut(L_2(17))\), \(L_2(25):2_3\), \(\Aut(U_3(3))\) or the dihedral group of order 38.
scientific article; zbMATH DE number 5354594
scientific article