On c-normal subgroups of some classes of finite groups

Abstract

Abstract A subgroup H is called c -normal in a group G if there exists a normal subgroup N of G such that HN = G and H ∩ N ≤ H G , where H G ≕ Core( H ) is the maximal normal subgroup of G which is contained in H . We obtain the c -normal subgroups in symmetric and dihedral groups. Also we find the number of c -normal subgroups of order 2 in symmetric groups. We conclude by giving a program in GAP for finding c -normal subgroups. AMS Classification : 20D25. Keywords : c -normal, symmetric, dihedral. Introduction The relationship between the properties of maximal subgroups of a finite group G and the structure of G has been studied extensively. The normality of subgroups in a finite group plays an important role in the study of finite groups. It is well known that a finite group G is nilpotent if and only if every maximal subgroup of G is normal in G . In Wang introduced the concept of c -normality of a finite group. He used the c -normality of a maximal subgroup to give some conditions for the solvability and supersolvability of a finite group. For example, he showed that G is solvable if and only if M is c -normal in G for every maximal subgroup M of G . In this paper, we obtain the c -normal subgroups in symmetric and dihedral groups, and also we find the number of c -normal subgroups of order 2 in symmetric groups.