Structure of Finite p-groups with Given Subgroups

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Structure of Finite p-groups with Given Subgroups

Journal Article•

Structure of Finite p-groups with Given Subgroups

01 Jan 2006-Contemporary mathematics-Vol. 402, pp 13-93

TL;DR: The following main results are proved: 1) Classification of p-groups all of whose subgroups of index p (square power) are abelian; 2) classification of p groups with 7 involutions; and 3) Classification of minimal nonmetacyclic p-group, classification of groups of odd order without normal elementary abelians of order p (power 3); and 4) Proofs of some basic counting theorems based on the new enumeration principle as discussed by the authors.

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Abstract: The following main results are proved: 1) Classification of p-groups all of whose subgroups of index p (square power)are abelian. 2) Classification of p-groups with 7 involutions. 3) New proofs of some Blackburn's results (classification of minimal nonmetacyclic p-groups, classification of p-groups of odd order without normal elementary abelian subgroup of order p (power 3)). 4) Proofs of some basic counting theorems based on the new enumeration principle. 5) New proof of Ward's theorem on quaternion-free 2-groups. 6) Corrected proof of Iwasawa's theorem on the structure of modular p-groups. Our proofs are completely elementary.

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Journal Article•DOI•

Noether's problem for p-groups with a cyclic subgroup of index p2

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Ming-chang Kang¹•Institutions (1)

National Taiwan University¹

15 Jan 2011-Advances in Mathematics

TL;DR: In this paper, it was shown that if G is a non-abelian p-group of order p n ( n ⩾ 3 ) containing a cyclic subgroup of index p 2 and K is any field containing a primitive p n − 2 -th root of unity, then G is rational over K.

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26 citations

Cites methods from "Structure of Finite p-groups with G..."

...In Section 3, we recall the classification of non-abelian pgroups with a cyclic subgroup of index p(2) by Ninomiya [Ni], which was reproved by Berkovich and Janko [BJ1]....
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...In Section 3, we recall the classification of non-abelian pgroups with a cyclic subgroup of index p2 by Ninomiya [Ni], which was reproved by Berkovich and Janko [BJ1]....
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...[BJ1] Y. Berkovich and Z. Janko, Structure of finite p-groups with given subgroups, in “Ischia Group Theory 2004”, edited by Z. Arad etc., Contemp....
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...A different proof of Ninomiya’s Theorem was given by Berkovich and Janko [BJ1, Section 11; BJ2, Section 74]....
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...[BJ2] Y. Berkovich and Z. Janko, Groups of prime power order, vol. 2, Walter de Gruyter, Berlin, 2008....
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Journal Article•DOI•

Finite p-Groups all of Whose Subgroups of Index p^3 are Abelian

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Qinhai Zhang¹, Libo Zhao¹, Miaomiao Li¹, Yiqun Shen¹•Institutions (1)

Shanxi Teachers University¹

04 Apr 2015

TL;DR: In this paper, the Frattini subgroup, the derived subgroup and the center of every \({\mathcal {A}}_3\)-group were classified up to isomorphism.

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Abstract: Suppose that \(G\) is a finite \(p\)-group. If all subgroups of index \(p^t\) of \(G\) are abelian and at least one subgroup of index \(p^{t-1}\) of \(G\) is not abelian, then \(G\) is called an \({\mathcal {A}}_t\)-group. We use \({\mathcal {A}}_0\)-group to denote an abelian group. From the definition, we know every finite non-abelian \(p\)-group can be regarded as an \({\mathcal {A}}_t\)-group for some positive integer \(t\). \({\mathcal {A}}_1\)-groups and \({\mathcal {A}}_2\)-groups have been classified. Classifying \({\mathcal {A}}_3\)-groups is an old problem. In this paper, some general properties about \({\mathcal {A}}_t\)-groups are given. \({\mathcal {A}}_3\)-groups are completely classified up to isomorphism. Moreover, we determine the Frattini subgroup, the derived subgroup and the center of every \({\mathcal {A}}_3\)-group, and give the number of \({\mathcal {A}}_1\)-subgroups and the triple \((\mu _0,\mu _1,\mu _2)\) of every \({\mathcal {A}}_3\)-group, where \(\mu _i\) denotes the number of \({\mathcal {A}}_i\)-subgroups of index \(p\) of \({\mathcal {A}}_3\)-groups.

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22 citations

Journal Article•DOI•

On finite nonabelian 2-groups all of whose minimal nonabelian subgroups are of exponent 4

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Zvonimir Janko¹•Institutions (1)

Heidelberg University¹

15 Sep 2007-Journal of Algebra

TL;DR: In this paper, Janko et al. showed that a minimal non-Dedekindian finite 2-group is either minimal nonabelian or is isomorphic to Q 16.

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17 citations

Journal Article•DOI•

Finite p-groups with a minimal non-abelian subgroup of index p (I)☆

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Haipeng Qu¹, Sushan Yang¹, Mingyao Xu¹, Lijian An¹•Institutions (1)

Shanxi Teachers University¹

15 May 2012-Journal of Algebra

TL;DR: For an odd prime p, this article classified finite p-groups with a unique minimal non-abelian subgroup of index p. In fact, such groups have a maximal quotient which is a 3-group of maximal class.

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17 citations

Cites background from "Structure of Finite p-groups with G..."

...Many leading group theorists, for example, Glauberman, Janko, etc., have turned their attentions to the study of finite p-groups....
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...Berkovich and Janko [3] introduced a new concept — At -groups, which is a more general concept than that of minimal nonabelian p-groups....
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...As Janko mentioned in the foreword of [4], to study p-groups with “large” abelian subgroups is another approach to finite p-groups....
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...Many scholars studied and classified A2-groups, see, for example [3,5,9,12,18,23]....
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...Berkovich and Janko [3] introduced a new concept —...
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Journal Article•DOI•

Finite 2-groups with exactly one nonmetacyclic maximal subgroup

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Zvonimir Janko¹•Institutions (1)

Heidelberg University¹

14 Aug 2008-Israel Journal of Mathematics

TL;DR: In this article, the authors determine the structure of the title groups in terms of generators and relations, and many important subgroups of these groups are described, where the minimal number of generators of a title group is defined.

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Abstract: We determine here the structure of the title groups. All such groups G will be given in terms of generators and relations, and many important subgroups of these groups will be described. Let d(G) be the minimal number of generators of G. We have here d(G) ≤ 3 and if d(G) = 3, then G′ is elementary abelian of order at most 4.

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15 citations