FINITE p-GROUPS WHOSE NON-CENTRAL CYCLIC SUBGROUPS HAVE CYCLIC QUOTIENT GROUPS IN THEIR CENTRALIZERS

Title & Authors
FINITE p-GROUPS WHOSE NON-CENTRAL CYCLIC SUBGROUPS HAVE CYCLIC QUOTIENT GROUPS IN THEIR CENTRALIZERS
Zhang, Lihua; Wang, Jiao; Qu, Haipeng;

Abstract
In this paper, we classified finite p-groups G such that $\small{C_G(x)/}$$\small{&}$$\small{lt;x}$$\small{&}$$\small{gt;}$ is cyclic for all non-central elements $\small{x{\in}G}$. This solved a problem proposed By Y. Berkovoch.
Keywords
centralizers;non-central elements;normal rank;p-groups of maximal class;
Language
English
Cited by
References
1.
Y. Berkovich, Groups of Prime Power Order, Volume 1, Walter de Gruyter, Berlin, 2008.

2.
Y. Berkovich and Z. Janko, Structure of finite p-groups with given subgroups, Contemp. Math. 402, 13-93, Amer. Math. Soc., Providence, RI, 2006.

3.
N. Blackburn, Generalizations of certain elementary theorems on p-groups, Proc. London Math. Soc. (3) 11 (1961), 1-22.

4.
B. Huppert, Endliche Gruppen I, Springer-Verlag, 1967.

5.
K. Ishikawa, Finite p-groups up to isoclinism, which have only two conjugacy lengths, J. Algebra 220 (1999), no. 1, 333-354.

6.
X. H. Li and J. Q. Zhang, Finite p-groups and centralizers of non-central elements, Comm. Algebra 41 (2013), no. 9, 3267-3276.

7.
M. Y. Xu, L. J. An, and Q. H. Zhang, Finite p-groups all of whose non-abelian proper subgroups are generated by two elements, J. Algebra 319 (2008), no. 9, 3603-3620.