Finite $p$-Groups with Large or Small Normal Closures of Non-Normal Cyclic Subgroups

Key Word: $JC$-group   non-Dedekindian $p$-group   regular $p$-group
Fund ProjectL:Supported by the National Natural Science Foundation of China (Grant Nos.11526114; 11601245), the Natural Science Foundation of Guangdong Province (Grant No.2015A030313791), the Innovative Team Project of Guangdong Province (CHINA) (Grant No.2014KTSCX196), Guangdong Province Innovation Talent Project for Youths (Grant No.2015KQNCX107) and the Appropriative Researching Fund for Doctors, Guangdong University of Education (Grant No.2013ARF07).
 Author Name Affiliation Libo ZHAO Department of Mathematics, Guangdong University of Education, Guangdong 510310, P. R. China Lv GONG School of Sciences, Nantong University, Jiangsu 226007, P. R. China
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A $p$-group $G$ is called a $JC$-group if the normal closure $H^G$ of every cyclic subgroup $H$ satisfies $|G:H^G|\leq p$ or $|H^G:H|\leq p$. In this paper, we classify the non-Dedekindian $JC$-groups for $p>2$.