Ordering groups and validity in lattice-ordered groups

An inductive characterization is given of the subsets of a group that extend to the positive cone of a right order on the group. This characterization is used to relate validity of equations in lattice-ordered groups (?-groups) to subsets of free groups that extend to the positive cone of a right order. As a consequence, new proofs are obtained of the decidability of the word problem for free ?-groups and generation of the variety of ?-groups by the ?-group of automorphisms of the real line. An inductive characterization is also given of the subsets of a group that extend to the positive cone of an order on the group. In this case, the characterization is used to relate validity of equations in varieties of representable ?-groups to subsets of relatively free groups that extend to the positive cone of an order.     

On the atomic conditions of lattice-ordered groups

We introduce large convex -subgroups to study the structure of the lattice-ordered groups G whose C(G), P(G) and (G) satisfy atomic conditions, where C(G), P(G) and (G) denote respectively the lattice of all convex -subgroups, the lattice of all polar subgroups and the root system of all regular subgroups of G. In particular, we construct a new torsion class defined as the class of -groups G for which all large prime subgroups are maximal. We prove that the class of hyperarchimedean -groups is properly contained within and that any -group within has the property that any chain of prime subgroups has length at most 2.Received October 7, 2003; accepted in final form June 11, 2004.     

Divisibility in O-primitive lattice-ordered permutation groups

We show that divisibility fails in a wide spectrum of periodic lattice-ordered permutation groups. The methods are applied to the six previously known ‘pathological’ doubly transitive groups represented on ℝ or on ℝ⁺, and only two of these groups turn out to be divisible.     

Varieties minimal over representable varieties of lattice-ordered groups

Weinberg showed that the variety of abelian lattice-ordered groups is the minimal nontrivial variety in the lattice of varieties of lattice-ordered groups. Scrimger showed that the abelian variety of lattice-ordered groups has countably infinitely many nonrepresentable covering varieties, and it is now known that his varieties are the only nonrepresentable covers of the abelian variety.

In this paper, a variation of the method used to construct the Scrimger varieties is developed that is shown to produce every nonrepresentable cover of any representable variety. Using this variation, all nonrepresentable covers of any weakly abelian l-variety are specifically identified, as are the nonrepresentable covers of any l-metabelian representable l-variety. In both instances, such il-varieties have only countably infinitely many such covers.

Any nonrepresentable cover of a representable il-variety is shown to be a subvariety of a quasi-representable il-variety as defined by Reilly. The class of these quasi-representable l-varieties is shown to contain the well-known L_n l-varieties and to generalize many of their properties.     

Normal subgroups and elementary theories of lattice-ordered groups

We show that any lattice-ordered group (l-group) G can be l-embedded into continuously many l-groups H _i which are pairwise elementarily inequivalent both as groups and as lattices with constant e. Our groups H _i can be distinguished by group-theoretical first-order properties which are induced by lattice-theoretically nice properties of their normal subgroup lattices. Moreover, they can be taken to be 2-transitive automorphism groups A(S _i ) of infinite linearly ordered sets (S _i , ) such that each group A(S _i ) has only inner automorphisms. We also show that any countable l-group G can be l-embedded into a countable l-group H whose normal subgroup lattice is isomorphic to the lattice of all ideals of the countable dense Boolean algebra B.     

A characterisation of virtually free groups

We prove that a finitely generated group G is virtually free if and only if there exists a generating set for G and k > 0 such that all k-locally geodesic words with respect to that generating set are geodesic. Received: 30 October 2006     

Free generating sets of lattice-ordered abelian groups

We prove that the generators g₁,…,g_n of a lattice-ordered abelian group G form a free generating set iff each ?-ideal generated by any n−1 linear combinations of the g_i is strictly contained in some maximal ?-ideal of G.     

An Abstract Cogalois Theory for profinite groups

The aim of this paper is to develop an abstract group theoretic framework for the Cogalois Theory of field extensions.     

Small conjugacy classes in the automorphism groups of relatively free groups

Let F be an infinitely generated free group and let R be a fully invariant subgroup of F such that (a) R is contained in the commutator subgroup F^′ of F and (b) the quotient group F/R is residually torsion-free nilpotent. Then the automorphism group of the group F/R^′ is complete. In particular, the automorphism group of any infinitely generated free solvable group of derived length at least two is complete.This extends a result by Dyer and Formanek (1977) [7] on finitely generated groups F_n/R^′ where F_n is a free group of finite rank n at least two and R a characteristic subgroup of F_n.     

The word problem in free normal valued lattice-ordered groups: a solution and practical shortcuts

In any lattice-ordered group (l-group) generated by a setX, every element can be written (not uniquely) in the form w(x)=⋁ _i ⋀ _j w _ij (x), where eachw _ij (x) is a group word in the elements ofX. An algorithm will be given for deciding whetherw(x) is the identitye in the free normal valuedl-group onX, or equivalently, whether the statement “∀x,w(x)=e” holds in all normal valuedl-groups. The algorithm is quite different from the one given recently by Holland and McCleary for the freel-group, and indeed the solvability of the word problem was established first for the normal valued case. The present algorithm makes crucial use of the fact (due to Glass, Holland, and McCleary) that the variety of normal valuedl-groups is generated by the finite wreath powersZ Wr Z Wr...Wr Z of the integersZ. In general, use of the algorithm requires a fairly large amount of work, but in several important special cases shortcuts are obtained which make the algorithm very quick. This is an expanded version of material developed while the author was on leave at Bowling Green State University in Bowling Green, Ohio, and presented in 1978 at the Conference on Ordered Groups at Boise State University in Boise, Idaho [9]. Presented by L. Fuchs.     

Comments on monoreflections in unital archimedean lattice-ordered groups and rings

In quite general concrete categories, the monoreflections which are H-closed (closed under homomorphic image) are the “naturally occuring” ones, since each is comprised of objects closed under some family of functorial implicit operations. In the specific categories under consideration here, we have earlier shown the class of H-closed monoreflections to be a set, indeed, relatively small. In spite, or because of that very few non-H-closed monoreflections have been exhibited, in fact, none in the categories considered here. We shall exhibit many and assuming no measurable cardinal, a proper class.     

On the Kurosh rank of the intersection of subgroups in free products of groups

The Kurosh rank rK(H) of a subgroup H of a free product of groups Gα, α∈I, is defined accordingly to the classic Kurosh subgroup theorem as the number of free factors of H. We prove that if H₁, H₂ are subgroups of and H₁, H₂ have finite Kurosh rank, then , where , q^∗ is the minimum of orders >2 of finite subgroups of groups Gα, α∈I, q^∗:=∞ if there are no such subgroups, and if q^∗=∞. In particular, if the factors Gα, α∈I, are torsion-free groups, then .     

A variety of lattice-ordered groups containing all representable covers of the abelian variety

A small variety of representable lattice-ordered groups is constructed, which contains all of the representable covers of the abelian variety.     

On lattice-ordered commutative semigroups

Decompositions of elements into intersections of primal elements and into intersections of p-components are studied in certain lattice-ordered commutative semigroups, by making use of the new development in commutative ideal theory without finiteness conditions, due to Fuchs-Heinzer-Olberding [7]. Several results concerning ideals can be phrased as theorems in abstract ideal theory.The intersections we consider are in general not irredundant, and the associated prime elements are not unique. However, one can establish a canonical intersection that is often irredundant with uniquely determined associated primes.     

Polyhedral groups and graph amalgamation products

A polyhedral group G is defined to be the orientation-preserving subgroup of a discrete reflection group acting on hyperbolic 3-space $\text{H}$³, and having a fundamental polyhedron of finite volume. A special presentation for G is obtained from the geometry of the polyhedron. This gives G the structure of a graph amalgamation product, and which, in some cases, splits as a free product with amalgamation. The simplest examples of polyhedral groups are the so-called tetrahedral groups. Other examples are given amongst the the groups PGL(2,O_m), where O_m is the ring of algebraic integers in the quadratic imaginary field $\text{Q(}\text{-m}\text{)}$, m>0.     

Abelian subgroup separability of certain generalized free products of free or finitely generated nilpotent groups

In this paper we prove that certain generalized free products of abelian subgroup separable groups, amalgamating an infinite cyclic subgroup, are abelian subgroup separable. Applying this, we derive that tree products of free groups or finitely generated nilpotent groups, amalgamating infinite cyclic subgroups, are abelian subgroup separable.     

Twisted second homology groups of the automorphism group of a free group

In this paper, we compute the second homology groups of the automorphism group of a free group with coefficients in the abelianization of the free group and its dual group except for the 2-torsion part, using combinatorial group theory.