Finite p-groups in which the intersection of nonnormal subgroup and center has bounded order

Abstract

Given a non-negative integer i, a finite p-group G is called an $MC(p^i)$-group if $|H\cap Z(G)|\le p^i$ for every nonnormal subgroup H of G. It is proved that the center of $MC(p^i)$-group has order $\le p^i$ or its derived subgroup is of order $\le p^{i+1}$. And the classification of MC(p)-groups is obtained.

KEYWORDS:

• Center
• Dedekind p-group
• nonnormal subgroup

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 20D15

Acknowledgments

The authors would like to thank the referee for his or her valuable suggestions and useful comments contributed to the final version of this paper.

Additional information

Funding

The research of the work was supported by the National Science Foundation of China (No. 12071092, No. 12101135), the major project of Basic and Applied Research (Natural Science) in Guangdong Province, China (No. 2017KZDXM058), the project of the Science and Technology Program of Guangzhou Municipality, China (No. 201804010088) and the Characteristic Innovation Project (Natural Science) of Guangdong Province (No. 2020KTSCX093).