Authors: BIJAN TAERI, FATEMEH TAYANLOO BEYG

Abstract: We characterize finite groups with exactly two nonabelian proper subgroups. When $G$ is nilpotent, we show that $G$ is either the direct product of a minimal nonabelian $p$-group and a cyclic $q$-group or a $2$-group. When $G$ is nonnilpotent supersolvable group, we obtain the presentation of $G$. Finally, when $G$ is nonsupersolvable, we show that $G$ is a semidirect product of a $p$-group and a cyclic group.

Keywords: Finite groups, minimal nonabelian groups, minimal nonnilpotent groups, critical groups

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