Hostname: page-component-745bb68f8f-mzp66 Total loading time: 0 Render date: 2025-02-05T22:19:52.494Z Has data issue: false hasContentIssue false
  • English
  • Français

ISOLATED SUBGROUPS OF FINITE p-GROUPS

Part of: Representation theory of groups

Published online by Cambridge University Press:  01 October 2021

QIANGWEI SONG  and
LIJIAN AN Open the ORCID record for LIJIAN AN [Opens in a new window]
QIANGWEI SONG
Affiliation:
Mathematics Department, Shanxi Normal University, Linfen, Shanxi 041004, China e-mail: [email protected]
LIJIAN AN*
Affiliation:
Mathematics Department, Shanxi Normal University, Linfen, Shanxi 041004, China
*
e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We say that a subgroup H is isolated in a group G if for each $x\in G$ either $x\in H$ or $\langle x\rangle \cap H={1}$ . We determine the structure of finite p-groups with isolated minimal nonabelian subgroups and finite p-groups with an isolated metacyclic subgroup.

Keywords

finite p-groupisolated subgroupmetacyclic subgroupHughes subgroup

MSC classification

Primary: 20D15: Nilpotent groups, $p$-groups
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 105 , Issue 3 , June 2022 , pp. 449 - 457
Check for updates
Copyright
© 2021 Australian Mathematical Publishing Association Inc.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

This work was supported by NSFC (Nos. 11901367 and 11971280).

References

An, L. J., Li, L. L., Qu, H. P. and Zhang, Q. H., ‘Finite $p$ -groups with a minimal nonabelian subgroup of index $p$ (II)’, Sci. China Ser. A 57 (2014), 737–753.CrossRefGoogle Scholar
Berkovich, Y., Groups of Prime Power Order I (Walter de Gruyter, Berlin–New York, 2008).Google Scholar
Hogan, G. T. and Kappe, W. P., ‘On the ${H}_p$ -problem for finite $p$ -groups’, Proc. Amer. Math. Soc. 20 (1969), 450–454.Google Scholar
Janko, Z., ‘Finite $p$ -groups with some isolated subgroups’, J. Algebra 465 (2016), 41–61.CrossRefGoogle Scholar
Qu, H. P., Yang, S. S., Xu, M. Y. and An, L. J., ‘Finite $p$ -groups with a minimal nonabelian subgroup of index $p$  (I)’, J. Algebra 358 (2012), 178–188.CrossRefGoogle Scholar
Xu, M. Y., An, L. J. and Zhang, Q. H., ‘Finite $p$ -groups all of whose nonabelian proper subgroups are generated by two elements’, J. Algebra 319 (2008), 3603–3620.Google Scholar
Zhang, Q. H., ‘Finite $p$ -groups all of whose minimal nonabelian subgroups are non-metacyclic of order ${p}^3$ ’, Acta Math. Sin. (Engl. Ser.) 35 (2019), 1179–1189.CrossRefGoogle Scholar