A Classification of Finite Metahamiltonian p-Groups

Abstract

A finite non-abelian group G is called metahamiltonian if every subgroup of G is either abelian or normal in G. If G is non-nilpotent, then the structure of G has been determined. If G is nilpotent, then the structure of G is determined by the structure of its Sylow subgroups. However, the classification of finite metahamiltonian p-groups is an unsolved problem. In this paper, finite metahamiltonian p-groups are completely classified up to isomorphism.

Keywords

Minimal non-abelian groups / Hamiltonian groups / Metahamiltonian groups / ${\mathcal {A}}_2$-groups')">${\mathcal {A}}_2$-groups

References

[1] An LJ. Finite $p$-groups whose non-normal subgroups have few orders. Front. Math. China. 2018, 13 4 763-777

[2] An LJ, Hu RF, Zhang QH. Finite $p$-groups with a minimal non-abelian subgroup of index $p$ (IV). J. Algebra Appl.. 2015, 14 2 1550020

[3] An LJ, Li LL, Qu HP, Zhang QH. Finite $p$-groups with a minimal non-abelian subgroup of index $p$ (II). Sci. China Math.. 2014, 57 737-753

[4] An LJ, Peng J. Finite $p$-groups in which any two noncommutative elements generate an inner abelian group of order $p^4$. Algebra Colloq.. 2013, 20 215-226

[5] An LJ, Zhang QH. Finite metahamilton $p$-groups. J. Algebra. 2015, 442 23-35

[6] Berkovich Y. Finite $p$-groups with few minimal nonabelian subgroups. J. Algebra. 2006, 297 62-100

[7] Berkovich Y, Janko Z. Structure of finite $p$-groups with given subgroups. Contemp. Math.. 2006, 402 13-93

[8] Berkovich Y, Janko Z. Groups of Prime Power Order. 2008 Berlin: Walter de Gruyter

[9] Blackburn N. On prime power groups with two generators. Proc. Camb. Philos. Soc.. 1957, 54 327-337

[10] Dedekind, R.: Über Gruppen, deren samtliche Teiler Normalteiler sind. Math. Ann. 48, 548–561 (1897)

[11] Draganyuk, S.V.: On the structure of finite primary groups all 2-maximal subgroups of which are abelian (Russian). In: Complex Analysis, Algebra and Topology. Akad. Nauk Ukrain. SSR, Inst. Mat., Kiev, pp. 42–51 (1990)

[12] Kazarin LS. On certain classes of finite groups. Dokl. Akad. Nauk SSSR (Russ.). 1971, 197 773-776

[13] King BW. Presentations of metacyclic groups. Bull. Austral. Math. Soc.. 1973, 8 103-131

[14] Nagrebeckii VT. Finite non-nilpotent groups any non-abelian subgroup of which is normal. Ural. Gos. Univ. Mat. Zap.. 1967, 6 80-88

[15] Newman MF, Xu MY. Metacyclic groups of prime-power order (research announcement). Adv. Math. (Beijing). 1988, 17 106-107

[16] Passman DS. Nonnormal subgroups of $p$-groups. J. Algebra. 1970, 15 352-370

[17] Qu HP, Yang SS, Xu MY, An LJ. Finite $p$-groups with a minimal non-abelian subgroup of index $p$ (I). J. Algebra. 2012, 358 178-188

[18] Qu HP, Xu MY, An LJ. Finite $p$-groups with a minimal non-abelian subgroup of index $p$ (III). Sci. China Math.. 2015, 58 763-780

[19] Qu HP, Zhao LP, Gao J, An LJ. Finite $p$-groups with a minimal non-abelian subgroup of index $p$ (V). J. Algebra Appl.. 2014, 13 7 1450032

[20] Rédei L. 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 gehören(German). Comment. Math. Helvet.. 1947, 20 225-264

[21] Sheriev VA. A description of the class of finite p-groups whose 2-maximal subgroups are all abelian II. Primary groups. Proc. Sem. Algebr. Syst.. 1970, 2 54-76

[22] Song QW, Qu HP. Finite $p$-groups whose nonnormal subgroups are metacyclic. Sci. China Math.. 2020, 63 7 1271-1284

[23] Song QW, Zhang QH. Finite $2$-groups whose length of nonnormal subgroups is at most 2. Front. Math. China. 2018, 13 5 1075-1097

[24] Xu MY, An LJ, Zhang QH. Finite $p$-groups all of whose non-abelian proper subgroups are generated by two elements. J. Algebra. 2008, 319 3603-3620

[25] Xu MY, Zhang QH. A classification of metacyclic $2$-groups. Algebra Colloq.. 2006, 13 25-34

[26] Zhang QH, Guo XQ, Qu HP, Xu MY. Finite groups which have many normal subgroups. J. Korean Math. Soc.. 2009, 46 1165-1178

[27] Zhang QH, Li XX, Su MJ. Finite $p$-groups whose nonnormal subgroups have orders at most $p^3$. Front. Math. China.. 2014, 9 5 1169-1194

[28] Zhang QH, Su MJ. Finite $2$-groups whose nonnormal subgroups have orders at most $2^3$. Front. Math. China. 2012, 7 5 971-1003

[29] Zhang QH, Sun XJ, An LJ, Xu MY. Finite $p$-groups all of whose subgroups of index $p^2$ are abelian. Algebra Colloq.. 2008, 15 167-180

[30] Zhang QH, Zhao LB, Li MM, Shen YQ. Finite $p$-groups all of whose subgroups of index $p^3$ are abelian. Commun. Math. Stat.. 2015, 3 1 69-162

Funding

National Natural Science Foundation of China(11971280)