Finite p-groups all of whose non-abelian proper subgroups are generated by two elements

In this paper, we give a complete classification of finite p-groups all of whose subgroups of index p2 are abelian.

Finite p-Groups All of Whose Subgroups of Index p2 Are Abelian

A characterization of metacirculants

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.

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

This paper finishes the classification of three-generator finite p -groups G such that Φ( G )≤ Z ( G ). This paper is a part of classification of finite p -groups with a minimal non-abelian subgroup of index p , and partly solves a problem proposed by Berkovich (2008).

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

A finite p-group G is called metacyclic if it has a cyclic normal subgroup A such that G/A is cyclic. In 1973, King classified metacyclic p-groups; in 1987, Newman and the first author developed a new approach to metacyclic p-groups suggested by the p-group generation algorithm, they found new presentations for these groups. However, their results were only announced in the case of p odd, and no proof was given. The purpose of this paper is to give a new proof of their results in the case of p=2, which is independent of the p-group generation algorithm.

A Classification of Metacyclic 2-Groups

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

Let G be a finite group. A subgroup H of G is called conjugate-permutable in G if HHg = HgH for any g ∈ G. A group G is called an ECP-group if every subgroup of G is conjugate-permutable in G. In this paper, we study the influence of conjugate-permutable subgroups on the structure of a finite group, especially on the nilpotency or supersolvability of the group, and give some sufficient or necessary conditions for a finite group to be an ECP-group.

Mingyao Xu

Papers

Finite p-groups all of whose non-abelian proper subgroups are generated by two elements

A Classification of Metacyclic 2-Groups

Finite p-Groups All of Whose Subgroups of Index p2 Are Abelian

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

On Conjugate-Permutable Subgroups of a Finite Group