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FINITE $p$-GROUPS ALL OF WHOSE NONNORMAL SUBGROUPS HAVE BOUNDED NORMAL CORES

Part of: Special aspects of infinite or finite groups Representation theory of groups

Published online by Cambridge University Press:  18 July 2019

DONGFANG YANG Open the ORCID record for DONGFANG YANG [Opens in a new window] ,
LIJIAN AN Open the ORCID record for LIJIAN AN [Opens in a new window]  and
HENG LV Open the ORCID record for HENG LV [Opens in a new window]
DONGFANG YANG*
Affiliation:
School of Mathematics and Statistics, Southwest University, Chongqing, 400715, PR China email [email protected]
LIJIAN AN
Affiliation:
Department of Mathematics, Shanxi Normal University, Linfen, Shanxi, 041004, PR China email [email protected]
HENG LV
Affiliation:
School of Mathematics and Statistics, Southwest University, Chongqing, 400715, PR China email [email protected]
*
✉[email protected]
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Abstract

Given a positive integer $m$, a finite $p$-group $G$ is called a $BC(p^{m})$-group if $|H_{G}|\leq p^{m}$ for every nonnormal subgroup $H$ of $G$, where $H_{G}$ is the normal core of $H$ in $G$. We show that $m+2$ is an upper bound for the nilpotent class of a finite $BC(p^{m})$-group and obtain a necessary and sufficient condition for a $p$-group to be of maximal class. We also classify the $BC(p)$-groups.

Keywords

normal corenon-Dedekindian groupmaximal class

MSC classification

Primary: 20D15: Nilpotent groups, $p$-groups
Secondary: 20F19: Generalizations of solvable and nilpotent groups
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 101 , Issue 2 , April 2020 , pp. 255 - 265
Copyright
© 2019 Australian Mathematical Publishing Association Inc.

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Footnotes

This work was supported by the National Natural Science Foundation of China (11671324, 11471266).

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