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 Finite $p$-groups whose non-central cyclic subgroups have cyclic quotient groups in their centralizers Bull. Korean Math. Soc. 2015 Vol. 52, No. 2, 367-376 https://doi.org/10.4134/BKMS.2015.52.2.367Published online March 31, 2015 Lihua Zhang, Jiao Wang, and Haipeng Qu Beijing University of Posts and Telecommunications, Shanxi Normal University, Shanxi Normal University Abstract : In this paper, we classified finite $p$-groups $G$ such that $$C_G(x)/\langle x\rangle$$ is cyclic for all non-central elements $x\in G$. This solved a problem proposed By Y. Berkovoch. Keywords : centralizers, non-central elements, normal rank, $p$-groups of maximal class MSC numbers : 20D15 Downloads: Full-text PDF