Finite Nonabelian Group Research Articles

The Definability of the Commutator Subgroup in a Variety Generated by a Finite Group

AbstractIn a variety generated by a finite non-Abelian group, the commutator subgroup is not definable by a first-order formula.

Finite simple non-Abelian groups

Finite simple non-Abelian groups

Two Undecidability Results using Modified Boolean Powers

In this paper we will give brief proofs of two results on the undecidability of a first-order theory using a construction which we call a modified Boolean power. Modified Boolean powers were introduced by Burris in late 1978, and the first results were announced in [2]. Subsequently we succeeded in using this construction to prove the results in this paper, namely Ershov's theorem that every variety of groups containing a finite non-abelian group has an undecidable theory, and Zamjatin's theorem that a variety of rings with unity which is not generated by finitely many finite fields has an undecidable theory. Later McKenzie further modified the construction mentioned above, and combined it with a variant of one of Zamjatin's constructions to prove the sweeping main result of [3]. The proofs given here have the advantage (over the original proofs) that they use a single construction.

Integral Group Rings of Some p-Groups

1. Introduction. The group of units, , of the integral group ring of a finite non-abelian group G is difficult to determine. For the symmetric group of order 6 and the dihedral group of order 8 this was done by Hughes-Pearson [3] and Polcino Milies [5] respectively. Allen and Hobby [1] have computed , where A4 is the alternating group on 4 letters. Recently, Passman-Smith [6] gave a nice characterization of where D2p is the dihedral group of order 2p and p is an odd prime. In an earlier paper [2] Galovich-Reiner-Ullom computed when G is a metacyclic group of order pq with p a prime and q a divisor of (p – 1). In this note, using the fibre product decomposition as in [2], we give a description of the units of the integral group rings of the two noncommutative groups of order p3, p an odd prime. In fact, for these groups we describe the components of ZG in the Wedderburn decomposition of QG.

Embedding group amalgams

A class of groups is said to have the small embedding property if every amalgam of two groups from amalgamating a normal subgroup embeds into a group not generating the variety of all groups. Very few large classes have the small embedding property. For example, the minimal non-abelian varieties containing non-abelian finite groups do not; neither does any class of groups containing ; but the set of finitely generated groups in a nilpotent variety not containing does have the small embedding property.

Groups with restricted non-normal subgroups

The study of the structure of groups all of whose subgroups have a prescribed property has been a consistent theme in the theory of groups. One of the earliest examples of work in this area is the paper of Miller-Moreno [12J which gives a classification of all finite non-Abelian groups in which every proper subgroup is Abelian (such groups, without the finiteness assumption, are called MillerMoreno groups). The structure of finite non-nilpotent groups with all proper subgroups nilpotent is taken up by Schmidt [23] and later by Gol'fond [7]. In a somewhat different line, Baer [1] gives a complete description of Dedekind groups (=groups with all subgroups normal). Baer's results extend to infinite groups the classical theorem of Dedekind [5]. Various combinations of the above ideas have also been studied. For example, Romalis and Sesekin [20], [21] and [22], study rneta-Hamiltonian groups these are groups with all subgroups normal or Abelian. In the above papers, Romalis and Sesekin obtain information primarily about locally solvable meta-Hamiltonian groups. Recent results show that the structure of Miller-Moreno groups will, in general, by very complex. For example, Ol'ghankii [16] has recently constructed a torsion-free simple group in which every proper subgroup is cyclic; Ol'~hanki~ also constructs infinite periodic groups in which every proper subgroup has prime order. In a parallel development, Rips [A generalization of small cancellation theory] has constructed infinite 2-generator groups of prime exponent in which every proper subgroup is cyclic. These remarkable examples suggest that Miller-Moreno groups (and hence meta-Hamiltonian groups) will admit a structure theory only in the presence of some finiteness condition. As a consequence of work on minimal conditions, Phillips and Wilson [17] have determined, under a finiteness condition, those groups in which every

Finite nonabelian subgroups of SU(n) with analytic expressions for the irreducible representations and the Clebsch–Gordan coefficients

We present two sequences of finite nonabelian groups which are semidirect products of three Zn groups. Although these groups are not simply reducible (in the tensor product of two irreducible representations an irreducible representation is obtained more than once) we give analytic expressions for the irreducible representations and the Clebsch–Gordan coefficients.

Self-duality conditions for spin systems defined on a new class of non-abelian solvable finite groups

A new class of finite non-abelian groups is presented. We work out explicitly the conditions under which the corresponding spin systems are self-dual.

Phase structure of non-Abelian lattice gauge theories

The phase structure of four-dimensional lattice gauge theories based on finite non-Abelian groups is studied by Monte Carlo computations. All models examined exhibit a two-phase structure with a first-order phase transition. In three systems where the gauge group is a discrete subgroup of SU(2) the critical temperature moves toward zero as the order of the group increases and the high-temperature phase has confining properties.

On the class groups of dihedral groups

Let A be a Z-order in a semisimple Q-algebra -4. Let C(A) denote the class group of A defined by using locally free left A-modules. Define D(d) to be the kernel of the natural map C(A) + C(Q) for Q a maximal Z-order in A containing A and d(A) to be the order of D(d). The integral group ring ZG of a finite group G is a Z-order in the semisimple Q-algebra QG, and hence the groups C(ZG) and D(ZG) can be defined. The study of the groups C(ZG) and D(ZG) seems to be important, because it has applications to various problems in number theory, algebraic topology, etc. In this paper we will try, as a continuation of [3], to determine finite nonabelian groups G such that d(ZG) = 1. Denote by D, the dihedral group of order 2n and by S, , A, the symmetric, alternating group on n symbols, respectively. The following theorem has been obtained in [3].

Growth sequences of finite groups IV

AbstractThe main result is that d(Ssm) = n+2 for every finite non-abelian two-generator simple group S of order s and every integer n > 0. This is applied to give a very close estimate on d(Gn) for any finite group G whose simple images are two-generator. The article is based on the author's previous papers with similar titles.

The near-rings hosted by a class of groups

Clay (3), Johnson (5) and Krimmel (6) have each considered the near-rings with identity on dihedral groups. Krimmel actually generalised the class of dihedral groups and investigated the class of finite non-abelian groups with a cyclic normal subgroup of prime index: we shall call this class . Krimmel considered the near-rings with identity that might be denned on members of and he determined the subclass of groups in which support near-rings of this kind. He also managed to calculate the number of non-isomorphic near-rings involved for certain cases. His methods were essentially combinatorial, and his results were expressed in terms of various integers which characterised the individual members of . Certain features of this work led us to investigate the structure of near-rings on members of from a more algebraic point of view and thereby to complete and extend Krimmel's programme. Part of the work in this paper formed the basis of the second author's thesis (7). We should like to thank Dr. J. Krimmel for permission to include some of his results, and Dr. J. Meldrum who detected an error in our original formulation of Theorem 7.1.

Augmentation quotients of some nonabelian finite groups

1. LetGbe a group,ZGits integral group ring and Δ = ΔGthe augmentation idealZGBy anaugmentation quotientofGwe mean any one of theZG-moduleswheren, r≥ 1. In recent years there has been a great deal of interest in determining the abelian group structure of the augmentation quotientsQn(G) =Qn,1(G) and(see (1, 2, 7, 8, 9, 12, 13, 14, 15)). Passi(8) has shown that in order to determineQn(G) andPn(G) for finiteGit is sufficient to assume thatGis ap-group. Passi(8, 9) and Singer(13, 14) have obtained information on the structure of these quotients for certain classes of abelianp-groups. However little seems to be known of a quantitative nature for nonabelian groups. In (2) Bachmann and Grünenfelder have proved the following qualitative result: ifGis a finite group then there exist natural numbersn0and π such thatQn(G) ≅Qn+π(G) for alln≥n0; ifGωis the nilpotent residual ofGandG/Gωhas classcthen π divides l.c.m. {1, 2, …,c}. There do not appear to be any examples in the literature of this periodic behaviour forc> 1. One of goals here is to present such examples. These examples will be from the class of finitep-groups in which the lower central series is anNp-series.

Finite non-Abelian groups with complemented non-Abelian subgroups

Finite non-Abelian groups with complemented non-Abelian subgroups

Fast Fourier Transforms on Finite Non-Abelian Groups

Recent works [1]-[9] were devoted to the properties and application of Fourier transforms over finite Abelian groups and fast Fourier transforms for calculation of the corresponding spectra. In this correspondence we describe Fourier transforms on finite non-Abelian groups and appropriate algorithms of fast Fourier transforms.

Graphical Regular Representations of Non-Abelian Groups, II

The present paper is a sequel to the previous paper bearing the same title by the same authors [3] and which will be hereafter designated by the bold-face Roman numeral I. Further results are obtained in determining whether a given finite non-abelian group G has a graphical regular representation. In particular, an affirmative answer will be given if (|G|, 6) = 1.Inasmuch as much of the machinery of I will be required in the proofs to be presented and a perusal of I is strongly recommended to set the stage and provide motivation for this paper, an independent and redundant introduction will be omitted in the interest of economy.

Lattice Isomorphisms of Finite Non-Abelian Groups of Exponent p

Lattice Isomorphisms of Finite Non-Abelian Groups of Exponent p

Lattice isomorphisms of finite non-abelian groups of exponent 𝑝.

Lattice isomorphisms of finite non-abelian groups of exponent 𝑝.