In this window, all groups are assumed finite. Here we collect a number of results that play a significant role in the book (further material of an elementary nature that we sometimes take for granted is easily available in textbooks such as [H], [R] and [A]).

Finite Group Theory

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)

Need a project to satisfy your craving for abstract algebra? Or do you have a student nosing around your office looking for a project? The Chermak–Delgado lattice is a sublattice of the subgroup la...

Exploring the Chermak-Delgado Lattice

In this note, we describe the structure of finite groups G whose Chermak–Delgado lattice is the interval [G∕Z(G)] = {H∈L(G)∣Z(G)≤H≤G}.

A note on the Chermak–Delgado lattice of a finite group

If G is a finite group with subgroup H, then the Chermak–Delgado measure of H (in G) is defined as |H||C G (H)|. The Chermak–Delgado lattice of G, denoted &#x1d49e;&#x1d49f;(G), is the set of all subgroups with maximal Chermak–Delgado measure; this set is a moduar sublattice within the subgroup lattice of G. In this paper we provide an example of a p-group P, for any prime p, where &#x1d49e;&#x1d49f;(P) is lattice isomorphic to 2 copies of ℳ2 (a quasiantichain of width 2) that are adjoined maximum-to-minimum. We introduce terminology to describe this structure, called a 2-string of 2-diamonds, and we also give two constructions for generalizing the example. The first generalization results in a p-group with Chermak–Delgado lattice that, for any positive integers n and l, is a 2l-string of n-dimensional cubes adjoined maximum-to-minimum and the second generalization gives a construction for a p-group with Chermak–Delgado lattice that is a 2l-string of ℳ p+1 (quasiantichains, each of width p + 1) adjoined maximum-to-minimum.

Chermak–Delgado Lattice Extension Theorems

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

In a finite group G with subgroup H, the Chermak-Delgado measure of H (in G) is defined as the product of the order of H with the order of the centralizer of H. The Chermak-Delgado lattice of G, denoted CD(G), is the set of all subgroups with maximal Chermak-Delgado measure; this set is a sublattice within the subgroup lattice of G. In this paper we provide an example of a p-group P, for any prime p, where CD(P) is lattice isomorphic to 2 copies of M_4 (a quasiantichain of width 2) that are adjoined maximum-to-minimum. We introduce terminology to describe this structure, called a 2-string of 2-diamonds, and we also give two constructions for generalizing the example. The first generalization results in a p-group with Chermak-Delgado lattice that, for any positive integers n and l, is a 2l-string of n-dimensional cubes adjoined maximum-to-minimum and the second generalization gives a construction for a p-group with Chermak-Delgado lattice that is a 2l-string of M_(p+3) (quasiantichains, each of width p + 1) adjoined maximum-to-minimum.

Chermak-Delgado Lattice Extension Theorems

The number of conjugacy classes of nonnormal subgroups of finite p-groups

In this paper, we classify finite 2-groups with a unique minimal non-abelian subgroup of index 2. We also classify finite 2-groups with a 2-generator abelian subgroup of index 2. This paper is a part of classification of finite p-groups with a minimal non-abelian subgroup of index p. Together with our previous work, we solved completely a problem proposed by Y. Berkovich.

Haipeng Qu

Papers

Chermak–Delgado Lattice Extension Theorems

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

Chermak-Delgado Lattice Extension Theorems

The number of conjugacy classes of nonnormal subgroups of finite p-groups

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