Research ArticleNo Access

Finite $𝒟 C$ -groups

Dandan Zhang

School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, P. R. China

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Haipeng Qu

School of Mathematics and Computer Science, Shanxi Normal University, Linfen Shanxi 041004, P. R. China

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, and

Yanfeng Luo

https://orcid.org/0000-0003-0513-334X

School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, P. R. China

E-mail Address: luoyf@lzu.edu.cn

Corresponding author.

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https://doi.org/10.1142/S0219498823500755Cited by:0 (Source: Crossref)

Abstract

Let G be a group and $DS (G) = {H^{'} | H \leq G}$ . G is said to be a $𝒟 C$ -group if $DS (G)$ is a chain under set inclusion. In this paper, we prove that a finite $𝒟 C$ -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 an optimal upper bound of $d (G^{'})$ for a $𝒟 C$ p-group G. We also prove that a $𝒟 C$ p-group is metabelian when $p \leq 3$ and provide an example showing that a non-abelian $𝒟 C$ p-group is not necessarily metabelian when $p \geq 5$ . In particular, $𝒟 C$ 2-groups are characterized.

Communicated by M. L. Lewis

Keywords:

AMSC: 20D15, 20D30, 20F05, 20F14

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