Finite DC-groups
Group Theory
2020-01-22 v1
Abstract
Let G be a group and DS(G) = { H'| H is any subgroup of G}. G is said to be a DC-group if DS(G) is a chain. In this paper, we prove that a finite DC-group is a semidirect product of a Sylow p-subgroup and an abelian p'-subgroup. For the case of G being a finite p-group, we obtain some properties of a DC-group. In particular, a DC 2-group is characterized. Moreover, we prove that DC-groups are metabelian for p<5 and give an example that a non-abelian DC-group is not be necessarily metabelian for p>3.
Keywords
Cite
@article{arxiv.2001.06832,
title = {Finite DC-groups},
author = {Dandan Zhang and Haipeng Qu and Yanfeng Luo},
journal= {arXiv preprint arXiv:2001.06832},
year = {2020}
}