Finite Groups with Few Almost Maximal Subgroups

Let G be a finite group. L. Morgana, C. E. Praeger, A.E. Zalesski {MOR} introduced the notion of almost maximal subgroup of G. A subgroup H of G is said to be almost maximal in G if NG(H) is maximal in G. In this paper, let $AM(G)$ denote the number of conjugacy classes of all almost maximal subgroups of G. First, we prove that G is Dedekindian if and only if $AM(G)=0$. Second, we show that the finite groups with AM(G)≤ 1 are supersolvable. Furthermore, we characterize the structure of finite groups with $AM(G)=1$. Final, we investigate a lower bound of $AM(G)$ for certain non-solvable groups.