Abstract
Assume p is an odd prime. We investigate finite p-groups all of whose minimal nonabelian subgroups are of order p3. Let \(\mathcal{P}_1\)-groups denote the p-groups all of whose minimal nonabelian subgroups are nonmetacyclic of order p3. In this paper, the \(\mathcal{P}_1\)-groups are classified, and as a by-product, we prove the Hughes’ conjecture is true for the \(\mathcal{P}_1\)-groups.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
An, L. J., Hu, R. F., Zhang, Q. H.: Finite p-groups with a minimal nonabelian subgroup of index p (IV). J. Algebra Appl., 14(2), 1550020 (54 pages) (2015)
An, L. J., Li, L. L., Qu, H. P., et al.: Finite p-groups with a minimal nonabelian subgroup of index p (II). Sci. China Ser. A, 57(4), 737–753 (2014)
An, L. J., Zhang, Q. H.: Finite metahamiltonian p-groups. J. Algebra, 442, 23–35 (2015)
Berkovich, Y.: Groups of Prime Power Order Vol. 1. Walter de Gruyter, Berlin. New York, 2008)
Berkovich, Y., Janko, Z.: Groups of Prime Power Order Vol. 2. Walter de Gruyter, Berlin. New York, 2008)
Berkovich, Y., Janko, Z.: Structure of finite p-groups with given subgroups. Contemp. Math., Amer. Math. Soc., Providence, RI, 402, 13–93 (2006)
Hogan, G. T., Kappe, W. P.: On the Hp-problem for finite p-groups. Proc. Amer. Math. Soc, 20, 450–454 (1969)
Hughes, D. R.: Research problem 3. Bull. Amer. Math. Soc., 63, 209 (1957)
Janko, Z.: Finite nonabelian 2-groups all of whose minimal nonabelian subgroups are of exponent 4. J. Algebra, 315, 801–808 (2007)
Qu, H. P., Xu, M. Y., An, L. J.: Finite p-groups with a minimal nonabelian subgroup of index p (III). Sci China Ser. A, 58(4), 763–780 (2015)
Qu, H. P., Yang, S. S., Xu, M. Y., et al.: Finite p-groups with a minimal nonabelian subgroup of index p (I). J. Algebra, 358, 178–188 (2012)
Qu, H. P., Zhao, L. P., Gao, J., et al.: Finite p-groups with a minimal nonabelian subgroup of index p (V). J. Algebra Appl., 13(7), 1450032 (35 pages) (2014)
Rédei, L.: Das schiefe Produkt in der Gruppentheorie. Comment. Math. Helvet., 20, 225–267 (1947)
Robinson, D. J. S.: A Course in the Theory of Groups, Second edition, Springer-Verlag, New York, 1996)
Wang, L. F., Zhang, Q. H.: Finite p-groups with a class of complemented normal subgroups. Acta. Math. Sinica, Engl. Ser., 33(2), 278–286 (2017)
Xu, M. Y., An, L. J., Zhang, Q. H.: Finite p-groups all of whose nonabelian proper subgroups are generated by two elements. J. Algebra, 319, 3603–3620 (2008)
Zhang, L. H., Qu, H. P.: At-groups satisfying a chain condition. J. Algebra Appl., 13(4), 1350137 (5 pages) (2014)
Zhang, L. H.: The intersection of nonabelian subgroups of finite p-groups. J. Algebra Appl., 16(1), 1750020 (9 pages) (2017)
Zhang, L. H.: The intersection of maximal subgroups which are not minimal nonabelian of finite p-groups. Comm. Algebra, 45(8), 3221–3230 (2017)
Zhang, Q. H., Sun, X. J., An, L. J., et al.: Finite p-groups all of whose subgroups of index p2 are abelian. Algebra Colloq., 15(1), 167–180 (2008)
Zhang, Q. H., Zhao, L. B., Li, M. M., et al.: Finite p-groups all of whose subgroups of index p3 are abelian. Commun. Math. Stat., 3(1), 69–162 (2015)
Zhao, L. B., Guo, X. Y.: Finite p-groups with exactly one A1-subgroup of given structure of order p 3. Acta. Math. Sinica, Engl. Ser., 29(11), 2099–2110 (2013)
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by National Natural Science Foundation of China (Grant Nos. 11771258 and 11471198)
Rights and permissions
About this article
Cite this article
Zhang, Q.H. Finite p-groups All of Whose Minimal Nonabelian Subgroups are Nonmetacyclic of Order p3. Acta. Math. Sin.-English Ser. 35, 1179–1189 (2019). https://doi.org/10.1007/s10114-019-7308-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10114-019-7308-x