Groups Of Finite Rank Research Articles

Abelian groups that are torsion over their endomorphism rings

Using Lambek torsion as the torsion theory, we investigate the question of when an Abelian group G is torsion as a module over its endomorphism ring E. Groups that are torsion modules in this sense are called ℒ-torsion. Among the classes of torsion and truly mixed Abelian groups, we are able to determine completely those groups that are ℒ-torsion. However, the case when G is torsion free is more complicated. Whereas no torsion-free group of finite rank is ℒ-torsion, we show that there are large classes of torsion-free groups of infinite rank that are ℒ-torsion. Nevertheless, meaningful definitive criteria for a torsion-free group to be ℒ-torsion have not been found.

On the Complexity of the Classification Problem for Torsion-Free Abelian Groups of Finite Rank

In this paper, we shall discuss some recent contributions to the project [15, 14, 2, 18, 22, 23] of explaining why no satisfactory system of complete invariants has yet been found for the torsion-free abelian groups of finite rank n ≥ 2. Recall that, up to isomorphism, the torsion-free abelian groups of rank n are exactly the additive subgroups of the n-dimensional vector space ℚn which contain n linearly independent elements. Thus the collection of torsion-free abelian groups of rank at most n can be naturally identified with the set S (ℚn) of all nontrivial additive subgroups of ℚn. In 1937, Baer [4] solved the classification problem for the class S(ℚ)of rank 1 groups as follows.Let ℙ be the set of primes. If G is a torsion-free abelian group and 0 ≠ x ϵ G, then the p-height of x is defined to behx(p) = sup{n ϵ ℕ ∣ There exists y ϵ G such that pny = x} ϵ ℕ ∪{∞}; and the characteristic χ (x) of x is defined to be the function

A question of B. Plotkin about the semigroup of endomorphisms of a free group

Let F F be a free group of finite rank n ≥ 2 n \geq 2 , let E n d ( F ) End(F) be the semigroup of endomorphisms of F F , and let A u t ( F ) Aut(F) be the group of automorphisms of F F . Theorem. If T : E n d ( F ) → E n d ( F ) T : End(F) \to End(F) is an automorphism of E n d ( F ) End(F) , then there is an α ∈ A u t ( F ) \alpha \in Aut(F) such that T ( β ) = α ∘ β ∘ α − 1 T(\beta ) = \alpha \circ \beta \circ \alpha ^{-1} for all β ∈ E n d ( F ) \beta \in End(F) .

Peterson-type dimension formulas for graded Lie superalgebras

Let be a free abelian group of finite rank, let Γ be a sub-semigroup of satisfying certain finiteness conditions, and let be a (Γ × Z2)-graded Lie superalgebra. In this paper, by applying formal differential operators and the Laplacian to the denominator identity of , we derive a new recursive formula for the dimensions of homogeneous subspaces of . When applied to generalized Kac-Moody superalgebras, our formula yields a generalization of Peterson’s root multiplicity formula. We also obtain a Freudenthal-type weight multiplicity formula for highest weight modules over generalized Kac-Moody superalgebras.

AUTOMORPHISMS OF RELATIVELY FREE GROUPS IN THE VARIETY N2A ∧ AN2 ∧ Nc

For positive integers n and c, with n [ges ] 2, let Gn, c be a relatively free group of finite rank n in the variety N2A ∧ AN2 ∧ Nc. It is shown that the subgroup of the automorphism group Aut(Gn, c) of Gn, c generated by the tame automorphisms and an explicitly described finite set of IA-automorphisms of Gn, c has finite index in Aut(Gn, c). Furthermore, it is proved that there are no non-trivial elements of Gn, c fixed by every tame automorphism of Gn, c.

Cohomology of Aut( Fn) in the p-rank two case

For odd primes p, we examine H ̂ ∗(Aut(F 2(p−1)); Z (p)) , the Farrell cohomology of the group of automorphisms of a free group F 2( p−1) on 2( P−1) generators, with coefficients in the integers localized at the prime (p)⊂ Z . This extends results by Glover and Mislin (J. Pure Appl. Algebra 150 (2) (2000)), whose calculations yield H ̂ ∗(Aut(F n); Z (p)) for n∈{ p−1, p} and is concurrent with work by Chen (Farrell cohomology of automorphism groups of free groups of finite rank, Ohio State University Ph.D. Dissertation, Columbus, Ohio, 1998) where he calculates H ̂ ∗(Aut(F n); Z (p)) for n∈{ p+1, p+2}. The main tools used are Ken Brown's “normalizer spectral sequence” (Brown, Cohomology of Groups, Springer, Berlin, 1982), a modification of Krstic and Vogtmann's (Comment. Math. Helv. 68 (1993) 216–262) proof of the contractibility of fixed point sets for outer space, and a modification of the Degree Theorem of Hatcher and Vogtmann (J. London Math. Soc. (2) 58 (3)(1998) 633–655).

Abelian Groups and Regular Modules

The structure of the additive group of a regular module is considered. Abelian groups that are regular modules over their rings of endomorphisms are studied. Nonreduced endoregular groups and endoregular torsion-free groups of finite rank are described.

On primitive representations of soluble groups of finite rank

In the paper it is proved, in particular, that a group is polycyclic if and only if it is soluble of finite rank, satisfies the ascending chain condition for normal subgroups and admits a faithful irreducible primitive representation over a field of characteristic zero. Methods are developed that enable one to study induced representations of nilpotent and soluble groups of finite rank.

On the Classification Problem for Rank 2 Torsion-Free Abelian Groups

We study here some foundational aspects of the classification problem for torsion-free abelian groups of finite rank. These are, up to isomorphism, the subgroups of the additive groups (Qn, +), for some n = 1, 2, 3,…. The torsion-free abelian groups of rank ⩽ n are the subgroups of (Qn, +).

Bounded Essential Extensions of Completely Decomposable Groups

Rank-one groups are torsion-free abelian groups isomorphic to some additive subgroup of the group of rational numbers. These groups have been classified by the so-called types that have a workable description. Completely decomposable groups are direct sums of rank-one groups and have been classified up to isomorphism. Almost completely decomposable groups are finite essential extensions of completely decomposable groups of finite rank. These groups have been studied extensively during the past 15 years and although much is known about them, it is a class of groups for which a complete understanding is beyond reach. In this paper we initiate the study of essential extensions of completely decomposable groups of arbitrary rank

Most automorphisms of a hyperbolic group have very simple dynamics

Let G be a non-elementary hyperbolic group (e.g. a non-abelian free group of finite rank). We show that, for “most” automorphisms α of G (in a precise sense), there exist distinct elements X +, X − in the Gromov boundary ∂G of G such that lim n→+∞ α ±n(g)=X ± for every g∈ G which is not periodic under α . This follows from the fact that the homeomorphism ∂α induced on ∂G has North–South (loxodromic) dynamics.

Catenarity in rings with abelian group action

Let R be a commutative ringG a free abelian group of finite rank n and R#G the corresponding skew group ring. We fix an integer m; 0 ≤ m ≤ n and a free subgroup Gm of G of rank m. We prove that if R is noetherian, if specG (R#Gm ) is (R,G)-normally separated and if the Laurent polynomial ring is catenary, then the ring R#Gm is G-catenary. In the particular case where R is an affine algebra over an algebraically closed field, we prove that if R is G-locally finite, then the skew group ring R#Gm is universally catenary and universally G-catenary. Furthermore Tauvel's height formula is valid in all prime factors and G-prime factors of R#Gm.

Direct Sums of Self-Small Mixed Groups

The class G of self-small mixed abelian groups of finite rank has recently been the focus of a number of investigations. It is known that, up to quasi-isomorphism, G is dual to the category of locally free torsion-free abelian groups of finite rank. We utilize the reduced mixed groups in G as building blocks of a more extensive class ∑G, the smallest class containing the reduced groups in G that is closed under taking infinite direct sums and direct summands. Our central result is the determination of a complete set of isomorphism invariants for the groups in ∑G. We supplement this broad classification theorem with an investigation of the fine structure of completely decomposable groups in ∑G.

Schreier theorem on groups which split over free abelian groups

Let G be either a free product with amalgamation A *c B or an HNN group A*c, where C is isomorphic to a free abelian group of finite rank. Suppose that both A and B have no nontrivial, finitely generated, normal subgroups of infinite indices. We show that if G contains a finitely generated normal subgroup N which is neither contained in C nor free, then the index of N in G is finite. Further, as an application of this result, we show that the fundamental group of a torus sum of 3-manifolds MI and M2, the interiors of which admit hyperbolic structures, have no nontrivial, finitely generated, nonfree, normal subgroup of infinite index if each of MI and M2 has at least one nontorus boundary.

Proficient presentations and direct products of finite groups

Let G be a finite group, F a free group of finite rank, R the kernel of a homomorphism φ of F onto G, and let [R, F], [R, R] denote mutual commutator subgroups. Conjugation in F yields a G-module structure on R/[R, R] let dg(R/[R, R]) be the number of elements required to generate this module. Define d(R/[R, F]) similarly. By an earlier result of the first author, for a fixed G, the difference dG(R/[R, R]) − d(R/[R, F]) is independent of the choice of F and φ; here it is called the proficiency gap of G. If this gap is 0, then G is said to be proficient. It has been more usual to consider dF(R), the number of elements required to generate R as normal subgroup of F: the group G has been called efficient if F and φ can be chosen so that dF(R) = dG(R/[R, F]). An efficient group is necessarily proficient; but (though usually expressed in different terms) the converse has been an open question for some time.

On a Theorem of Zelmanov

Our object here is to give a new proof of a theorem of Zelmanov [7] on pro-p groups having presentations with few relators and, while doing this, to prove an additional result which illustrates that these groups are large. We begin by making formal the notion of a presentation with few relators. Let F be a free pro-p group of finite rank, with Zassenhaus filtration Fi‘. (The definition of the Zassenhaus filtration is recalled below.) The degree ζg‘ of a non-identity element g of F is the integer i such that g ∈ Fi \Fi+1. For a subset U of F we define σU; i = Ž”u ∈ U Ž ζu‘ = i•Ž for each i ∈ , and if each σU; i is finite we define a real power series σUt‘ by

The conjugacy problem for Dehn twist automorphisms of free groups

A Dehn twist automorphism of a group G is an automorphism which can be given (as specified below) in terms of a graph-of-groups decomposition of G with infinite cyclic edge groups. The classic example is that of an automorphism of the fundamental group of a surface which is induced by a Dehn twist homeomorphism of the surface. For $ G = F_n $ , a non-abelian free group of finite rank n, a normal form for Dehn twist is developed, and it is shown that this can be used to solve the conjugacy problem for Dehn twist automorphisms of $ F_n $ .

The Generating Function for the Number of Roots of a Coxeter Group

Using elementary roots and finite automata, we show that the generating function counting the depths of the roots of a Coxeter group of finite rank is rational.

Duality and self-reflexive torsion-free abelian groups

We characterize the torsion-free abelian groups G of finite rank such that Hom(−, G) defines a rank preserving duality on the category of End( G)-submodules of finite sums of copies of G. Our results provide a maximal extension of Warfield's duality result for torsion-free groups of finite rank. In order to obtain our extension, we study the torsion-free abelian groups G for which G ≅ nat Hom(Hom( G, G), G).

Induced modules over group algebras of metabelian groups of finite rank

Induced modules over group algebras of metabelian groups of finite rank