On Minimal Non-Soluble Groups, the Normalizer Condition and McLain Groups

Abstract

We prove that a minimal non-soluble ($MN\mathfrak{S}$ in short) Fitting $p$-group $G$ has a proper subgroup $K$ such that for every proper subgroup $R$ of $G$ containing $K$, we have $N_G(R) > R$. In other words, $G$ satisfies the normalizer condition modulo $K$. We also give a positive answer in McLain groups to a question aroused from the works on $MN\mathfrak{S}$ Fitting $p$-groups.

Keywords

• McLain group
• normalizer condition
• minimal non-soluble group