Abstract
We classify finite p-groups all of whose nonnormal subgroups have orders at most p 3, p odd prime. Together with a known result, we completely solved Problem 2279 proposed by Y. Berkovich and Z. Janko in Groups of Prime Power Order, Vol. 3.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
An L, Li L, Qu H, Zhang Q. Finite p-groups with a minimal non-abelian subgroup of index p (II). Sci China Ser A, 2014, 57(4): 737–753
Berkovich Y. Short proofs of some basic characterization theorems of finite p-groups theory. Glas Mat Ser III, 2006, 41: 239–258
Berkovich Y. Groups of Prime Power Order, Vol 1. Berlin: Walter de Gruyter, 2008
Berkovich Y, Janko Z. Groups of Prime Power Order, Vol 2. Berlin: Walter de Gruyter, 2008
Berkovich Y, Janko Z. Groups of Prime Power Order, Vol 3. Berlin: Walter de Gruyter, 2011
Huppert B. Endliche Gruppen I. New York: Springer, 1967
James R. The groups of order p6 (p an odd prime). Math Comp, 1980, 34: 613–637
Passman D. Nonnormal subgroups of p-groups. J Algebra, 1970, 15: 352–370
Xu M. A theorem on metabelian p-groups and some consequences. Chin Ann Math Ser B, 1984, 5: 1–6
Xu M, An L, Zhang Q. Finite p-groups all of whose non-abelian proper subgroups are generated by two elements. J Algebra, 2008, 319: 3603–3620
Zhang Q, Guo X, Qu H, Xu M. Finite group which have many normal subgroups. J Korean Math Soc, 2009, 46(6): 1165–1178
Zhang Q, Su M. Finite 2-groups whose nonnormal subgroups have orders at most 23. Front Math China, 2012, 7(5): 971–1003
Zhang Q, Sun X, An L, Xu M. Finite p-groups all of whose subgroups of index p2 are abelian. Algebra Colloq, 2008, 15(1): 167–180
Zhang Q, Zhao L, Li M, Shen Y. Finite p-groups all of whose subgroups of index p3 are abelian (in preparation)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zhang, Q., Li, X. & Su, M. Finite p-groups whose nonnormal subgroups have orders at most p 3 . Front. Math. China 9, 1169–1194 (2014). https://doi.org/10.1007/s11464-014-0389-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11464-014-0389-z