On groups occurring as absolute centers of finite groups

Abstract

Given a construction f on groups, we say that a group G is f -realisable if there is a group H such that $G\cong f(H)$, and completely f-realisable if there is a group H such that $G\cong f(H)$ and every subgroup of G is isomorphic to $f(H_1)$ for some subgroup $H_1$ of H and vice versa.

Denote by $L(G)$ the absolute center of a group G, that is the set of elements of G fixed by all automorphisms of G. By using the structure of the automorphism group of a ZM-group, in this paper we prove that cyclic groups $C_N$, $N\in {\mathbb{N}}^{*}$, are completely L-realisable.

KEYWORDS:

• Absolute center of a group
• automorphism group
• (completely) f-realisable group
• inverse group theory
• ZM-group

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 20D30
• Secondary: 20D45
• 20D25

Notes

1 We note that in this case m is a cyclic number.

2 More precisely, since $\text{ZM}(5,48,2)\cong C_3\times \text{ZM}(5,16,2)$, we have $L(\text{ZM}(5,48,2))\cong L(C_3)\times L(\text{ZM}(5,16,2))\cong 1\times C_4\cong C_4$ and $Z(\text{ZM}(5,48,2))\cong Z(C_3)\times Z(\text{ZM}(5,16,2))\cong C_3\times C_4\cong C_{12}$.