논문KCI 등재
Finite $p$-groups all of whose subgroups of class 2 are generated by two elements
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We proved that finite $p$-groups in the title coincide with finite $p$-groups all of whose non-abelian subgroups are generated by two elements. Based on the result, finite $p$-groups all of whose subgroups of class 2 are minimal non-abelian (of the same order) are classified, respectively. Thus two questions posed by Berkovich are solved.
Abstract
We proved that finite $p$-groups in the title coincide with finite $p$-groups all of whose non-abelian subgroups are generated by two elements. Based on the result, finite $p$-groups all of whose subgroups of class 2 are minimal non-abelian (of the same order) are classified, respectively. Thus two questions posed by Berkovich are solved.
참고문헌 (12)
- Y. Berkovich, "Groups of Prime Power Order. Vol. 1" (2008), Walter de Gruyter GmbH & Co
- Y. Berkovich, "Groups of Prime Power Order. Vol. 2" (2008), Walter de Gruyter GmbH & Co
- Y. Berkovich, "Groups of Prime Power Order. Vol. 3" (2011), Walter de Gruyter GmbH & Co
- N. Blackburn, "Generalizations of certain elementary theorems on p-groups", Proc. Lond. Math. Soc, 11(3), pp.1~22 (1961)
- T. J. Laffey, "The minimum number of generators of a nite p-group", Bull. Lond. Math. Soc, 5, pp.288~290 (1973)
- L. Redei, "Das "schiefe Produkt" in der Gruppentheorie mit Anwendung auf die endlichen nichtkommutativen Gruppen mit lauter kommutativen echten Untergruppen und die Ordnungszahlen, zu denen nur kommutative Gruppen gehoren", Comment. Math. Helv, 20, pp.225~264 (1947)
- M. Y. Xu, "A theorem on metabelian p-groups and some consequences", Chin. Ann. Math. Ser. B, 5(1), pp.1~6 (1984)
- M. Y. Xu, "Finite p-groups all of whose non-abelian proper subgroups are generated by two elements", J. Algebra, 319(9), pp.3603~3620 (2008)
- M. Y. Xu, "Finite p-groups" (2010), Beijing University Press
- M. Y. Xu, "A classification of metacyclic 2-groups", Algebra Colloq, 13(1), pp.25~34 (2006)
- Q. H. Zhang, "Finite p-groups all of whose subgroups of index p2 are abelian", Algebra Colloq, 15(1), pp.167~180 (2008)
- Q. H. Zhang, "Finite p-groups all of whose subgroups of index p3 are abelian", Commun. Math. Stat, 3(1), pp.69~162 (2015)
관련 논문AI추천
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