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 ≤ HG, where HG ≕ 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.