General Group Research Articles

On groups with a permutational property on commutators of weight 4

We show that a periodic 2-generator group G with no involutions is metabelian if and only if for any $ a, x_1, x_2,x_3 \in G $ there exists a nontrivial permutation $ \sigma \in \Bbb {S}_3 $ such that $ [a, x_1, x_2, x_3 ] = [a,x_{\sigma (1)},x_{\sigma (2)},x_{\sigma (3)}] $ .

THIN GROUPS OF PRIME-POWER ORDER AND THIN LIE ALGEBRAS

IN this paper we extend the theory of thin groups of prime-power order in a direction suggested by the theory of groups of prime-power order with fixed co-class, [9,14]. Moreover, we have found it helpful to use associated Lie algebras in this study. Let p be a prime. Recall that a group of p-power order is thin [2] if all the anti-chains in its lattice of normal subgroups are short. Here an anti-chain in a lattice is a subset of pairwise incomparable elements; and by saying that anti-chains are short we mean that they have length at most p +1. In fact, the definition extends to pro-/? -groups, and their lattices of (closed) normal subgroups. Thus p -groups of maximal class, or co-class 1, are examples of thin groups; for the theory of groups of co-class 1 see [10]. Examples of thin infinite pro-p-groups are the binary p-adic group, [7, 111.17], and the Nottingham group [9,15]. Brandl, Caranti and Scoppola [3] have shown that metabelian thin p-groups have order at most p?. It is clear that the lattice of normal subgroups of a non-(pro-)cyclic (pro-)p-group has anti-chains of length p +1 . To avoid trivialities, it is convenient to exclude (pro-)cyclic groups from our consideration of thin groups here. The only thin abelian p-group is thus elementary of order p; its lattice of normal subgroups we refer to as a diamond. It follows that the commutator factor group of a thin group is elementary of order p, so that thin groups are 2-generator groups, and every factor of the lower central series has exponent p and order at most p. It also follows (see [2,3] and the next section) that every non-trivial normal (closed) subgroup in a thin group lies between two consecutive terms of the lower central series so that in a thin pro-p-group all normal subgroups are of finite index, that is, open. Thus the lattice of normal subgroups of a thin group looks like a sequence of diamonds linked by chains. The groups of

The Nonabelian Tensor Square and Schur Multiplicator of Nilpotent Groups of Class 2

The nonabelian tensor square and the Schur multiplicator are determined for arbitrary groups of class 2 in a closed form. A functorial description is given in terms of a polynomial quotient of the integral group ring, as well as a more explicit formula for finite groups which can be evaluated by matrix calculus. This is carried out for 2-generator groups. Also some homotopical data of the suspended classifying space of class 2 groups are derived. The approach is based on the use of polynomial maps and constructions in the sense of Passi.

Quasi-identities of torsion-free 2-generated groups

Quasi-identities of torsion-free 2-generated groups

Properly discontinuous actions on Λ-trees

The main result is a theorem giving several possibilities for the action of a 2-generator group acting on a Λ-tree, generalising the result that, if the action is free then the group is either free or free abelian. This involves investigation of several cases in which the action is shown to be properly discontinuous. This leads to a generalisation of results of Culler and Morgan, characterising abelian, dihedral and irreducible actions on ℝ-trees, to arbitrary Λ-trees.

Embedding Countable Groups in 2-Generator Groups

The aim of this note is to popularize a simple proof, due to Neumann and Neumann [5], of the fact that every countable group is embeddable in a 2-generator group. This was first proved by Higman, Neumann, and Neumann [2, Theorem IV] and (independently) Freudenthal [2, p. 254], using free products with amalgamations. The proof given by Neumann and Neumann [5] used wreath products, which are widely regarded as no less terrifying than free products with amalgamations. I am indebted to Professors A. M. W. Glass, G. Higman, and P. M. Neumann, each of whom pointed out (in response to an earlier version of this note) that the Neumann-Neumann proof is really quite simple, and that it can easily be expressed directly in terms of permutations. Here, then, is a short proof that assumes no more background than is ileeded to understand the statement of the theorem. As usual, Z is the set of integers and NJ is the set of natural numbers; (a, b> is the group generated by a and b; Sym(fQ) is the group of all permutations of a set Ql; permutations are regarded as right operators, and are composed from left to right.

Growth sequences of 2-generator simple groups

SynopsisA study is made of the minimum number of generators of the n-th direct power of certain 2-generator groups.

ON GROUPS ALL OF WHOSE PROPER SUBGROUPS ARE FINITE CYCLIC

For any odd number n≥1003, the authors construct an infinite 2-generator group each of whose proper subgroups is contained in a cyclic subgroup of order n. This result strengthens analogous results of Ol'shanskiĭ for prime n>1075 and Atabekyan and Ivanov for odd n>1080. The proof is carried out in the original language of Novikov-Adyan theory. Bibliography: 6 titles.

Imbedding of countable periodic groups into simple 2-generated periodic groups

We prove a theorem on the isomorphic imbedding of an arbitrary countable periodic group H into a simple 2-generated periodic group G. In addition, we show that for any integers k ≥ 2 and l ≥ 3 the group G contains a pair of generating elements whose orders are k and l.

Paradoxical decompositions, 2-generator Kleinian groups, and volumes of hyperbolic 3-manifolds

Paradoxical decompositions, 2-generator Kleinian groups, and volumes of hyperbolic 3-manifolds

A class of deficiency zero soluble groups of derived length 4

SynopsisIn this paper we study a class of 2-generator 2-relator groups G(m) and show that they are all finite. Moreover, two infinite subclasses are soluble of derived length 4.

Investigation on the Besm-6 computer of the structure of the nilpotent series for the metabelian 2-generated group of exponent 27

A special symbolic form for the notation of the elements of a 2-generated metabelian group is realized on a computer, with the aid of which for the exponent 27 one carries out the search of the relations in the factors of the lower central series. The first 36 factors are investigated.

Finite groups of deficiency zero involving the Lucas numbers

In this paper, we investigate a class of 2-generator 2-relator groups G(n) related to the Fibonacci groups F(2,n), each of the groups in this new class also being defined by a single parameter n, though here n can take negative, as well as positive, values. If n is odd, we show that G(n) is a finite soluble group of derived length 2 (if n is coprime to 3) or 3 (otherwise), and order |2n(n + 2)gnf(n, 3)|, where fn is the Fibonacci number defined by f0=0,f1=1,fn+2=fn+fn+1 and gn is the Lucas number defined by g0 = 2, g1 = 1, gn+2 = gn + gn+1 for n≧0. On the other hand, if n is even then, with three exceptions, namely the cases n = 2,4 or –4, G(n) is infinite; the groups G(2), G(4) and G(–4) have orders 16, 240 and 80 respectively.

Nontrivial identities in finitely generated semigroups

3-34 (1959). 7. G. E. Wall, "On Hughes' Hp-problem," in: Proc. Intern. Conf. Theory of Groups, Canberra, 1965, Gordon and Breach, New York (1967), pp. 357-362. 8. E. I. Khukhro, "On connection between Hughes' conjecture and relations in free groups of prime period," Mat. Sb., 116, No. 2, 253-264 (1981). 9. E. I. Khukhro, "On the Lie ring of a free 2-generated group of prime period and on Hughes' conjecture for 2-generated p-groups," Mat. Sb., 118, No. 4, 567-575 (1982). I0. E. I. Khukhro, "A new identity in the Lie ring of a free group of prime period and groups without Hughes' property," Izv. Akad. Nauk SSSR, Set. Mat., 50, No. 6, 1308-1325 (1986). ii. G. E. Wall, "On the multilinear identities which hold in the Lie ring of a group of prime-power exponent," J. Algebra, 104, 1-22 (1986). 12. J. R. McMullen, "Vortical pro-p-groups and the Hughes' conjecture," Preprint, Univ. of Sydney (1985). 13. R. Warfield, Nilpotent Groups, Lect. Notes in Math., Vol. 517, Springer-Verlag, Berlin (1976). 14. O. H. Kegel, "Die Nilpotenz der Hp-Gruppen," Math. Z., 75, 373-376 (1961).

Q-theories of 2-generated groups with given axiomatic rank

Q-theories of 2-generated groups with given axiomatic rank

Some special 2-generator Kleinian groups

We explore the following question. Let G G be the subgroup of PSL ( 2 , C ) {\text {PSL}}(2,{\mathbf {C}}) generated by the elements A A and B B , where A A has two fixed points, and B B maps one fixed point of A A onto the other; when is G G discrete?

A FINITELY GENERATED COMPLETE GROUP

An example of a 2-generated complete group (different from the identity) with unique extraction of roots is constructed. An example is indicated, in passing, of a noncyclic 2-generated group in which every element is conjugate to some power of a fixed element. It is proved that there are continuum many nonisomorphic such examples. The proof is based on a method developed in several papers of A. Yu. Ol'shanskiĭ. Bibliography: 8 titles.

Superimprimitive 2-generator finite groups

In his thesis, A. A. Hussein Omar, motivated by the study of possible shapes of generic Dirichlet regions for a surface group, made a detailed study for g = 2,3 of the groups generated by pairs (μ, τ) of regular (i.e. fixed-point-free) permutations of order 2,3 respectively and of degree n = 6(2g − 1), such that μ ْ τ is an n-cycle. He observed that, for g = 2,3, precisely one pair generates what he calls a superimprimitive group, and raised the question whether such pairs exist for all g, and, if so, whether they areunique. Our main result is that they do always exist, but that, for large values of g, theyare far from unique. (For details and some motivation for the notation, see [4, 5].)

Factors of the nilpotent series of certain metabelian groups of primary exponent

By using an additive technique proposed earlier by the author, defining relations in the factors of the nilpotent series of the maximal metabelian 2-generated groups of exponent p2 are found. A comparison of the upper bound 2p2−3 thus obtained for the nilpotency class with the exact bound 2 (p2−p), obtained in the paper abstracted in RZMat, 1969, 3A180, shows that the list of relations obtained is not complete. Further, it is shown as to how, by increasing the precision of computations, one succeeds in achieving a definitive description of such groups for the exponents 9 and 8.

On Multiplication Groups of Quasigroups

1.1. In [13] Smith raised the question as to which groups are isomorphic to multiplication groups of quasigroups. It will be shown that all finite dihedral, symmetric, alternating, general linear and projective general linear groups and the Mathieu-group M 11 have this property (Smith proved in [13] that the same is true for the Mathieu-group M 23). These results are proved using a theorem in which multiplication groups of quasigroups are characterized by relating them to multiplication groups of loops. However, in all these cases, except for the alternating groups of even degree, it suffices to use a corollary of this theorem. This corollary states that a permutation group is the multiplication group of a quasigroup if it is generated by three elements, one of them being a cyclic permutation of all symbols. This applies in particular to all 2-generated permutation groups which contain such a cyclic permutation. 1.2. A pair ( Q, · ), consisting of a set Q and a binary operation · on Q, is called a quasigroup if, for all x E Q, the right multiplication mapping R.(x): Q ~ Q, y~ y · x, and the left multiplication mapping L.(x): Q~ Q, y~x · y, are permutations of Q. Therefore on every quasigroup ( Q, ·) two additional binary operations I and can be defined by xl y := xR.(y)1 and x\y := yL.(x)• Then the following equations are satisfied (cf. Bruck [5]):