Restricted Wreath Product Research Articles

ON GROUPS WHICH ARE SYNTACTIC MONOIDS OF DETERMINISTIC CONTEXT-FREE LANGUAGES

The author shows that the class of groups which are syntactic monoids of deterministic context-free languages is closed under restricted standard wreath products with virtually free top groups. This provides an answer to a question of T. Herbst.

Finite generation and presentability of wreath products of monoids

This paper gives necessary and sufficient conditions for the restricted wreath product of two monoids to be finitely generated or finitely presented.

Wreath operations in the group of automorphisms of the binary tree

A new operation called tree-wreathing is defined on groups of automorphisms of the binary tree. Given a countable residually finite 2-group H and a free abelian group K of finite rank r this operation produces uniformly copies of these as automorphism groups of the binary tree such that the group generated by them is an over-group of the restricted wreath product H≀K . Indeed, G contains a normal subgroup N which is an infinite direct sum of copies of the derived group H ′ and the quotient group G / N is isomorphic to H≀K . The tree-wreathing construction preserves the properties of solvability, torsion-freeness and of having finite state (i.e., generated by finite automata). A faithful representation of any free metabelian group of finite rank is obtained as a finite-state group of automorphisms of the binary tree.

ON FINITE GENERATION AND OTHER FINITENESS CONDITIONS FOR WREATH PRODUCTS OF SEMIGROUPS

ABSTRACT The purpose of the paper is to investigate finite generation and some other basic finiteness conditions for (restricted) wreath products of semigroups (with respect to an idempotent ). The main result is that such a wreath product , with finite, is finitely generated if and only if , , is finitely generated and either is a finitely generated -act, or else every element of belongs to the principal right ideal of a right identity. Further results are obtained for the case where is infinite, and also for finite presentability, periodicity and local finiteness.

R̄-Groups

We introduce a new class of torsion-free groups, calledR̄-groups, in which the normalizer and the centralizer of the isolator of each cyclic subgroup coincide. This is an extensive class of groups containing all torsion-free locally nilpotent groups and all free groups. These groups share many interesting properties with the class of allR-groups, groups having unique roots. For example, we show that the class ofR̄-groups is closed under products, subgroups, restricted wreath products, free products, and certain free products with amalgamation. Because of these nice properties, the class induces well behaved closure operators and we investigate the concomitant categorically compact classes associated with these operators. These new compact classes are closed under finite products and homomorphic images and their characterization generally involves the existence of roots of all orders for elements of the hypercenter.

On the uniqueness of wreath products

Suppose R is a group and P is a permutation group acting on the set I. Let M be the restricted direct product of 111 copies of R. Then the restricted wreath product G = R wr( P, I) is the semidirect product of M and P where P acts on M by permuting the factors. We are interested in whether or not M, referred to as the base subgroup of G, is a characteristic subgroup of G. In the special case when P acts regularly on Z, i.e., we may take Z= P with P acting by right multiplication, then P. M. Neumann [9] showed that A4 is not a characteristic subgroup of G if, and only if, 1 PI = 2 and R is a special dihedral group (this will be defined in the next section). In [2], Yu. V. Bodnarchuk claimed that if P is transitive and consists entirely of linitary permutations (i.e., every element of P moves only finitely many points of I), and if M is not a characteristic subgroup of G, then P has a nonidentity normal elementary abelian 2-subgroup and R is of dihedral type (to be defined in the next section). This claim is incorrect and counterexamples (pointed out to me by L. G. Kovacs) using a construction of P. Hall [4] are given in Section 6. Bodnarchuk apparently overlooked the possibility that an automorphism of an infinite group can map a subgroup into a proper subgroup of itself.’ In the counterexamples to Bodnarchuk’s result, the distinct images of the base subgroup under all the automorphisms of G constitute an infinite totally ordered set under inclusion. Bodnarchuk’s argument is valid, however, for finite groups. More generally, we show that Bodnarchuk’s result becomes a correct theorem if

Growth series of some wreath products

The growth series of certain finitely generated groups which are wreath products are investigated. These growth series are intimately related to the traveling salesman problem on certain graphs. A large class of these growth series is shown to consist of irrational algebraic functions. n=0 Since only one generating set will be associated with each group below, the generating set associated with fr(x) will be obvious. Let H be a group with a finite generating set Sh ■ These will be fixed for the rest of the paper. The Cayley graph of the pair (H, Sh) is, as usual, the directed graph whose vertices are the elements of H and there is an edge from a vertex hx Xoa vertex h2 if and only if h2 = hxh for some h in SH US^1. In particular, the edge from hx to h2 has an opposite edge from h2 to hx . Let C be the graph gotten from the Cayley graph of (H, Sh) simply by identifying opposite edges. In other words, C might be called the undirected Cayley graph of (H,SH). hex K also be a group with a finite generating set Sk ■ These will also be fixed for the rest of the paper. It is possible to form what might be called a restricted direct product group P, which consists of all functions p from the vertices of C to K such that there are only finitely many vertices v in C with p(v) t? 1. This is a subgroup of the direct product group whose elements consist of all functions from the vertex set of C to K. The group P admits H as a group of automorphisms by means of the action of H on C. The resulting semidirect product P x H is the restricted wreath product K\H.

QUASIVARIETIES OF GROUPS CLOSED WITH RESPECT TO RESTRICTED WREATH PRODUCTS

This paper studies quasivarieties of groups, closed under restricted wreath products. It is shown that if a class of groups is closed with respect to restricted wreath products, then the quasivariety generated by is also closed under restricted wreath products. The base rank of a nontrivial quasivariety, closed under restricted wreath products, is found to be two. Conditions are given that ensure that a countable group from a given quasivariety is isomorphically embeddable in a 2-generator group from the same quasivariety. Finally, the cardinality of the set of all quasivarieties that consist of torsionfree groups and are closed with respect to restricted wreath products is computed.Bibliography: 12 titles.

On the Conjugacy Problem for F/R �

Let FIR be a finitely generated recursively presented group with the solvable conjugacy problem and the solvable power problem. The conjugacy problem for F/R' when F/R is torsion free has been solved by Remeslennikov and Sokolov (Algebra and Logic 9 (1970), 342-349). In this note we prove that F/R' has the solvable conjugacy problem even when F/R is not torsion free. This allows us, in particular, to conclude the solvability of the conjugacy problem for F/[Fl, FI] in terms of the conjugacy problem for F/Fn. Summary. If the conjugacy problem and the power problem for F/R is solvable, then the conjugacy problem for F/R' is solvable. Introduction. Let F/R be a finitely generated recursively presented group with the solvable conjugacy problem and the solvable power problem. The conjugacy problem for F/R' when F/R is torsion free has been solved by Remeslennikov and Sokolov [6]. In this note we are able to prove that F/R' has the solvable conjugacy problem even when F/R is not torsion free. This allows us, in particular, to conclude the solvability of the conjugacy problem for F/[Fn, Fn] in terms of the conjugacy problem for F/FI. As in [6] our tool is the Magnus embedding of F/R' into a group of 2 X 2 matrices and a result of Matthews [5] about the conjugacy problem for the restricted wreath product of two groups. Pre iminary results. Let F be a noncyclic free group on a finite set X and R a normal subgroup of F. Let Ql be the free left Z(F/R)-module with basis {X$jx E X}. Then every element of Q2 is of the form fZE fxX fx E Z(F/R). xX For w E F, we set Rw = w and consider the group M F/R Q2 O RIR) of 2 X 2 matrices of the form (w f 0O 1) w E F, f E ?2. It is easily verified that M is isomorphic to the restricted wreath product of a free abelian group of rank IXI by F/R. Thus, by a result of Matthews [5], we have Received by the editors October 6, 1980. 1980 Mathematics Subject Classification. Primary 20F10. lResearch supported by the Natural Sciences and Engineering Research Council Canada. i) 1982 American Mathematical Society 0002-9939/81/0000-1066/$02.25 149 This content downloaded from 157.55.39.111 on Wed, 03 Aug 2016 05:17:24 UTC All use subject to http://about.jstor.org/terms

A note on the normalizer condition

The first example of a non-trivial group satisfying the normalizer condition but having trivial centre was obtained by Heineken and Mohamed (1). In fact, the group they constructed satisfies the stronger condition that all its proper subgroups are subnormal and nilpotent. In this note we use a construction suggested by their approach to obtain such a group G as a subgroup of the restricted wreath product Cp ≀ Cp∞. The fact that G is given as a subgroup of a well known group makes it rather easier to investigate its properties, and in particular to see that its centre Z(G) is trivial.

On the semisimplicity of modular group algebras

Let G be a discrete group, let K be a field and let KG denote the group algebra of G over K. We say that KG is semisimple if its Jacobson radical JKG is zero. If K has characteristic 0 and K is not an algebraic extension of the rationals then it is known [l, Theorem 1 ] that KG is semisimple. Moreover it appears likely that in the remaining characteristic 0 cases we also have semisimplicity. Thus nothing particularly interesting occurs here. If K has characteristic p>0 and G is a p'-group (that is, G has no elements of order p), then it is known (see [2]) that KG has no nil ideals and that for suitably big fields the group algebra KG is semisimple. Again it appears that for the remaining fields we also have semisimplicity. The interest in characteristic p stems from the fact that, unlike the case in which G is finite, it is quite possible for G to have elements of order p and yet have the group algebra KG be semisimple. Several examples of such groups have been exhibited and in each case a big abelian group is involved as either a subgroup or a factor group. In this paper we study the group algebras KG of those groups G having a big abelian subgroup or factor group. The methods used are extremely elementary. Two interesting examples of the type of groups which we can deal with are as follows. Let P be a cyclic group of order p and let C be an infinite cyclic group. Let Gi = C P and G2=P I C where I denotes the restricted Wreath product. Now Gi has a normal torsion free abelian subgroup of finite index p and G2 has a normal elementary abelian p-subgroup E with G2/E infinite cyclic. Surprising as it may seem if K is an algebraically closed field of characteristic p, then 7<Gi and KG2 are both semisimple. For the remainder of this paper K is an algebraically closed field of characteristic p. By a linear 7I-character of KG we mean a FJ-homomorphism X: KG—*K.

Central series in wreath products

In this paper we study the structure of the α-central series of the nilpotent wreath product of two Abelian groups, the α-central series being the intersection of the upper central series with the base group. Let C = A wr B, the standard restricted wreath product of A and B. Then Baumslag(1) showed that C is nilpotent if and only if A is a nilpotent p-group of finite exponent and B is a finite p-group for the same prime p. In (5) Liebeck calculated the nilpotency class of the nilpotent wreath product of two Abelian groups. We obtain an expression for an element of the base group to belong to a given term of the upper central series.