Finite groups with three nonabelian subgroups

Abstract

We characterize finite groups with exactly two non-abelian proper subgroups. When G is nilpotent, we show that G is either the direct product of a minimal non-abelian p -group and a cyclic q -group or a 2 -group. When G is non-nilpotent supersolvable group, we obtain the presentation of G . Finally, when G is non-supersolvable, we show that G is a semi direct product of a p -group and a cyclic group.

Generalization of infinite groups

“…The group problem is also the same, and the content of infinite groups and finite groups is also completely different. In the history of mathematics for hundreds of years, it has been relatively full of limited research [8], but wireless things are relatively lacking in any subject, and the same is true of group theory. Among them, the aspect of non-isomorphic groups with the same set of elements orders has been blank for many years.…”