Finite nilpotent groups coincide with their $2$-closures in all of their faithful permutation representations

TL;DR

This paper characterizes finite nilpotent groups that are 2-closed in all faithful permutation representations, showing they are either cyclic or a product involving a generalized quaternion group.

Contribution

It provides a complete classification of finite nilpotent groups that are 2-closed in all faithful permutation representations, linking group structure to 2-closure properties.

Findings

Finite nilpotent groups are 2-closed iff they are cyclic or a product of a generalized quaternion group with an odd cyclic group.

Characterization of 2-closed groups in terms of their algebraic structure.

Establishes a clear criterion for 2-closure in finite nilpotent groups.

Abstract

Here we show that a finite nilpotent group is 2-closed if and only if it is either cyclic or a direct product of a generalized quaternion group with a cyclic group of odd order.