Brendan Goldsmith

ALGEBRAIC ENTROPY FOR ABELIAN GROUPS

The theory of endomorphism rings of algebraic structures allows, in a natural way, a systematic approach based on the notion of entropy borrowed from dynamical systems. Here we study the algebraic entropy of the endomorphisms of Abelian groups, introduced in 1965 by Adler, Konheim and McAndrew. The so-called Addition Theorem is proved; this expresses the algebraic entropy of an endomorphism φ of a torsion group as the sum of the algebraic entropies of the restriction to a φ-invariant subgroup and of the endomorphism induced on the quotient group. Particular attention is paid to endomorphisms with zero algebraic entropy as well as to groups all of whose endomorphisms have zero algebraic entropy. The significance of this class arises from the fact that any group not in this class can be shown to have endomorphisms of infinite algebraic entropy, and we also investigate such groups. A uniqueness theorem for the algebraic entropy of endomorphisms of torsion Abelian groups is proved.

The kaplansky test problems — an approach via radicals

Abstract The existence of non-free, k-free Abelian groups and modules (over some non-left perfect rings R) having prescribed endomorphism algebra is established within ZFC + ◊ set theory. The principal technique used exploits free resolutions of non-free R-modules X and is similar to that used previously by Griffith and Eklof; much stronger results than have been obtained heretofore are obtained by coding additional information into the module X. As a consequence we can show, inter alia, that the Kaplansky Test Problems have negative answers for strongly ℵ 1 - free Abelian groups of cardinality ℵ 1 in ZFC and assuming the weak Continuum Hypothesis.

TRANSITIVE AND FULLY TRANSITIVE GROUPS

The notions of transitivity and full transitivity for abelian pgroups were introduced by Kaplansky in the 1950s. Important classes of transitive and fully transitive p-groups were discovered by Hill, among others. Since a 1976 paper by Corner, it has been known that the two properties are independent of one another. We examine how the formation of direct sums of p-groups affects transitivity and full transitivity. In so doing, we uncover a far-reaching class of p-groups for which transitivity and full transitivity are equivalent. This result sheds light on the relationship between the two properties for all p-groups.

On Adjoint Entropy of Abelian Groups

The theory of endomorphism rings of algebraic structures allows, in a natural way, a systematic approach based on the notion of entropy borrowed from dynamical systems. In the present work we introduce a ‘dual’ notion based upon the replacement of the finite groups used in the definition of algebraic entropy, by subgroups of finite index. The basic properties of this new entropy are established, and a connection to Hopfian groups is investigated.

TORSION-FREE WEAKLY TRANSITIVE ABELIAN GROUPS

ABSTRACT We introduce the notion of weak transitivity for torsion-free abelian groups. A torsion-free abelian group G is called weakly transitive if for any pair of elements x, y ∈ G and endomorphisms ϕ, ψ ∈ End(G) such that xϕ = y, yψ = x, there exists an automorphism of G mapping x onto y. It is shown that every suitable ring can be realized as the endomorphism ring of a weakly transitive torsion-free abelian group, and we characterize up to a number-theoretical property the separable weakly transitive torsion-free abelian groups.

On quasi-permutation representations of finite groups

by J. M. BURNS, B. GOLDSMITH, B. HARTLEY and R. SANDLING(Received 5 January, 1993)1. Introduction. In [6], Wong defined a quasi-permutation group of degree n to bea finite group G of automorphisms of an n -dimensional complex vector space such thatevery element of G has non-negative integral trace. The terminology derives from the factthat if G is a finite group of permutations of a set Q of size n, and we think of G as actingon the complex vector space with basis £2, then the trace of an element geGis equal tothe number of points of Q fixed by g. In [6] and [7], Wong studied the extent to whichsome facts about permutation groups generalize to the quasi-permutation group situation.Here we investigate further the analogy between permutation groups and quasi-permutation groups by studying the relation between the minimal degree of a faithfulpermutation representation of a given finite group G and the minimal degree of a faithfulquasi-permutation representation. We shall often prefer to work over the rational fieldrather than the complex field.By a quasi-permutation matrix we mean a square matrix over the complex field Cwith non-negative integral trace. Thus, every permutation matrix over C is a quasi-permutation matrix. For a given finite group G, let p(G) denote the minimal degree of afaithful permutation representation of G (or of a faithful representation of G bypermutation matrices), let q(G) denote the minimal degree of a faithful representation ofG by quasi-permutation matrices over the rational field Q, and let c(G) be the minimaldegree of a faithful representation of G by complex quasi-permutation matrices. Thus,It is easy to see that if G is cyclic of order 6, then = c(G) q(G) = 4 and p{G) = 5, while onthe other hand, if G is the quaternion group of order 8, then c{G) = 4 and q{G) =p(G) = 8. Thus, both inequalities can be strict. It is not too hard to see that for the groupSL(2,5), both inequalities are strict (see Section 4). Our principal aim in this paper is toinvestigate these quantities and inequalities further, and our main theorem characterizesthose finite abelian groups for which equality holds.

On endomorphism rings of non-separable abelian p-groups

There have been spectacular advances in recent years in the so-called realization problem for certain classes of abelian groups and modules. The advances have derived mainly from powerful combinatorial set-theoretic tools pioneered by Shelah. In the case of separable abelian p-groups these results appear, e.g., in [ 1,4, 51. For non-separable abelian p-groups the most significant contribution has been [2]. Our objective in this paper is to extend the results in [2] and derive analogous results to those obtained in [4, 51. It is perhaps worth pointing out that our methods are entirely algebraic; the necessary set-theoretical work has been carried out in the separable case and no further set-theoretical arguments are needed. In this the work is reminiscent of [S]. All our set theory (with one exception) is in ZFC and this is not acknowledged in the statement of individual results. In the exceptional case where we work in Godel’s Constructible Universe (V= L) this is acknowledged by appending (V= L) after the statement number. The principal realization result can be stated as:

On socle-regularity and some notions of transitivity for Abelian

In the present work the interconnections between various notions of transitivity for Abelian p-groups and the recently introduced concepts of socle-regular and strongly socle-regular groups are studied.

p

Direct sum decompositions problems for torsion-free modules of nite rank have been the subject of recent activity in the theory of modules over valuation domains (see e.g. [11]). Indeed the nal problem (Problem 26) in the text by Fuchs and Salce [6] asks if direct decompositions of nite rank modules (over a valuation domain) into indecomposable summands are unique up to isomorphism i.e. does the Krull– Schmidt Theorem hold for this class of modules? It has been known for some time that the Krull–Schmidt Theorem fails for nite-rank torsion-free modules over the discrete valuation domain Zp, the integers localized at the prime p; see [1, Exercise 2.15] for an example due essentially to M.C.R. Butler. However, there is no immediate method of extending this result to arbitrary valuation domains; indeed it is well known that if a valuation domain is Henselian, then the Krull–Schmidt Theorem does hold for torsionfree modules of nite rank (see e.g. [11, Lemma 14] or [10, Corollary 10]. In fact for discrete rank one valuation domains the two concepts are equivalent [11, Theorem 17]. One of the principal outcomes of the present work is that the Krull–Schmidt Theorem fails for a large class of non-Henselian valuation domains. It is worth remarking that this class contains many valuation domains which are not discrete and so is a farreaching generalization of [11, Theorem 17]. (We also note that Facchini has recently shown failure of Krull–Schmidt for serial modules [5].) In order to make this comment a little more precise, let us introduce the following notation: throughout, unless speci ed to the contrary, R shall denote a valuation domain, not a eld, with completion R

-groups

An abelian group is said to be minimal if it is isomorphic to all its subgroups of finite index. We obtain a complete characterization of such groups in the torsion case; in the case of mixed groups of rank 1 we obtain a characterization for some large classes of such groups.