Criteria for Total Projectivity

The study of effectiveness in classical mathematics is rapidly expanding, through recent research in algebra, topology, model theory, and functional analysis. Well-known contributors are Barwise (Wisconsin), Crossley (Monash), Dekker (Rutgers), Ershoff (Novosibirsk), Feferman (Stanford), Harrington (Berkeley), Mal′cev (Novosibirsk), Morley (Cornell), Nerode (Cornell), Rabin (Hebrew University), Shore (Cornell). Further interesting work is due to Kalantari (University of California, Santa Barbara), Metakides (Rochester), Millar (Wisconsin), Remmel (University of California, San Diego), Nurtazin (Novosibirsk). Areas investigated include enumerated algebras, models of complete theories, vector spaces, fields, orderings, Hilbert spaces, and boolean algebras.We investigate the effective content of the structure theory of p-groups. Recall that a p-group is a torsion abelian group in which the (finite) order of each element is some power of a fixed prime p. (In the sequel, “group” = “additively written abelian group”.)The structure theory of p-groups is based on the two elementary notions of order and height. Recall that the order of x is the least integer n such that nx = 0. The height of x is the number of times p divides x, that is, the least n such that x = pny for some y in the group but x ≠ pn+1y for any y. If for each n ∈ ω there is a “pnth-root” yn, so that x = pnyn, then we say that x has infinite height. In 1923, Prüfer related the two notions as criteria for direct sum decomposition, provingTheorem. Every group of bounded order is a direct sum of cyclic groups, andTheorem. Every countable primary group with no (nonzero) elements of infinite height is a direct sum of cyclic groups.

The study of Hopficity in Abelian groups has been largely motivated by the fundamental results of Baumslag, who proved that torsion groups are always Hopfian regardless of their cardinality, but left several questions open concerning torsion-free groups. Later, Corner addressed some of these questions by providing counterexamples showing that a direct sum of two Hopfian groups can be non-Hopfian, and that a group with an automorphism group of order two does not guarantee Hopficity. These results highlighted the need for new constructions to explore Hopficity in torsion-free Abelian groups. Our work introduces a new approach based on divisibility techniques, as it contributes to the understanding of free-torsion groups with respect to the Hopficity property, providing new insights into their structural properties and implications within group theory. Our analysis also demonstrates how divisibility properties, as well as the introduction of totally invariant subgroups and homomorphisms, can be used to establish Hopficity in specific Abelian groups, particularly those that are free-torsion. In order to reach all of this, we start by taking a group defined as an infinite direct sum of cyclic groups; then we construct a specific subgroup generated by two particular families of elements; and finally we show that this group is Hopfian through results from the theory of divisible subgroups.

The fundamental theorem of abelian groups states that any finitely generated abelian group is a direct sum of cyclic groups. This theorem plays a fundamental role in the structure theory of abelian groups. It has fascinated many algebraists to look at this theorem from different points of view:- (a) To find suitable generalizations of this theorem for modules over certain classes of rings, e.g. Dedekind domain, hereditary noetherian prime rings, valuation rings etc. (b) To create suitable versions of this theorem in some module categories and use such versions to develop the structure theory of such module categories. For example, torsion abelian group-like modules, modules with finitely generated submodules direct sums of multiplication modules. (c) To find those rings for which certain versions of the fundamental theorem of abelian groups hold. For example rings over which all finitely generated modules are direct sums of cyclic modules. (d) To examine the structure of certain classes of abelian groups and to try to find modules with similar structures. (e) To find roles of answers to some of above types of questions in the general theory of modules.

On the computational complexity of the theory of Abelian groups

CHAPTER V - BASIC SUBGROUPS

Introduction.Given a field k and a fc-algebra A, Gerstenhaber has introduced the concept of a deformation of A [8] which is a [[/]]-algebra structure on the module A [[/]].One of the main problems of the theory of deformations is to determine those algebras for which the only deformation (modulo an equivalence relation) is the usual A-[[r]]-algebra structure on .4[[*]].Such algebras are said to be rigid.If A is a commutative /c-algebra, we may restrict our attention to those deformations of A which are commutative.This leads us to consider the commutative cohomology groups of A, Hn(A, M)s, defined by Harrison [11].The principal purpose of this paper is to investigate the following conditions on a commutative algebra A over a field k:(1) H2(A, M)s = 0 for all ^-modules M.(2) H\A,f=0.(3) A is a rigid /r-algebra in the commutative deformation theory.(4) The module of differentials of A over k is a projective /4-module.(5) A is a finite product of (not necessarily algebraic) separable extension fields of k.We may show that (1) implies (4) [9, Theorem (20.4.9)] and that (2) implies (3) [8, p. 65].Condition (1) may be expressed by saying that A is a formally smooth fc-algebra [9, 19.4.4].When A is a field, Gerstenhaber has shown that (1), (2), and ( 5) are equivalent and he conjectured that these conditions were equivalent to (3).We show that (1), ( 2), (3), and (5) are equivalent even for A semisimple (Theorem 6.5).Moreover, if A is artinian, then (1) and ( 5) are equivalent (Proposition 5.9).In another direction, if A is a geometric complete intersection, then (1), (2), and (3) are equivalent (Theorem 6.8a).If in addition A is an integral domain with a separable quotient field, then (1), ( 2), (3), and (4) are equivalent (Theorem 6.8c).In the final section, we investigate the above conditions for the group algebra B[G] of an abelian group G which is a direct sum of cyclic groups with B a commutative algebra over a field k of positive characteristic p.In addition, we consider the following condition on G and k:(6) G has no element of order p = char (k).

For a given family (Gi)i∈N of finitely generated abelian groups, we construct a Dedekind domain D having the following properties.(1)Pic(D)≅⨁i∈NGi.(2)For each i∈N, there exists a submonoid Si⊆D• with Pic(DSi)≅Gi.(3)Each class of Pic(D) and of all Pic(DSi) contains infinitely many prime ideals. Furthermore, we study orders as well as sets of lengths in the Dedekind domain D and in all its localizations DSi.

Suppose that G is a p-primary abelian group. The subgroup G[p] = {x∈G:px=0} is called the socle of G and any subgroup S of G[p] is called a subsocle of G. If each subsocle of G supports a pure subgroup, then G is said to be pure-complete [1]. It is well known that, if G a direct sum of cyclic groups, then G is necessarily pure-complete. Further results about pure-complete groups are contained in [1] and [3].

A cotorsion hull of the separable 𝑝-group 𝑇 is considered when 𝑇 is a direct sum of torsion-complete groups. It is proved that in the considered case its cotorsion hull is fully transitive if and only if 𝑇 is a direct sum of cyclic groups or is a torsion-complete group.

Necessary conditions are given in order that Tor(Λ, B) be a direct sum of cyclic groups for abelian groups A and B. Sufficient conditions are also given that compare with the necessary conditions.They are only slightly stronger if at all, and the two are equivalent for groups of cardinality not exceeding K 2 * Due to the complexity of the problem, these conditions are not absolute, but constitute a reduction to smaller cardinality.The results generalize earlier results of R. Nunke.What is known of the structure and properties of Tor(^4, B) for abelian groups A and B is primarily due to R. Nunke.The main results appear in [4], [5], and [6].Contributions of other authors, however, were manifested in [2] and [3].This note generalizes the results of Nunke published in [4] and [6] concerning a basic question: When is Tor(^4, B) a direct sum of cyclic groups?Nunke has provided a completely satisfactory answer when neither the cardinality of A nor B exceeds 8j and in certain other cases, as well, allowing A and B to be arbitrarily large.However, the general case where A or B has cardinality greater than # λ was deferred.In this paper, we settle the question for cardinality X 2 .For the general case, we give necessary conditions in order that Ύoτ{A, B) be a direct sum of cyclic groups, and \γe also give sufficient conditions that are very close to the necessary ones, but the gap is not bridged completely.Since Tor(^4, B) is not affected by the nontorsion portions of A and 5, we can assume without loss of generality that A and B are torsion.Further, we can specialize, as usual, to the case that A and B are both /7-primary.Therefore, all groups are assumed to be/^-primary.The following three theorems summarize the major known results concerning the question of when Tor(^4, B) is a direct sum of cyclic groups.Following [6], we say that G is Σ-cyclic if G is a direct sum of cyclic groups.THEOREM A {Nunke [4, Corollary 3.5]).If p ω A φ 0, then Tor(^, B) is Σ-cyclic only if B is Έ-cyclic.The preceding result also appears in [6] as Theorem 12(i).

Primary decomposition of the J-groups of complex projective and lens spaces

We shall characterize the complete first order theories of torsion free abelian groups having algebraically prime models, elementarily prime models, minimal models, etc., using the well known invariants from [9], [3]. This study was suggested by the questions raised in [2]. Group here will always mean abelian group. For reference on groups see Fuchs [5], [6], for model theory Chang-Keisler [1]. All theories considered are theories in the first order language of group theory. A group G is minimal (strictly minimal) if H < G implies H = G (H-= G), and elementarily prime (algebraically prime) if G is elementarily (algebraically) embeddable in every H G. p, q always denote prime numbers, P the set of all primes. The p-rank of a group G, p-rk(G), is the dimension of G/pG as a vector space over Z/pZ. We define tf(p, G): = min {p-rk(G), ~o}. By the results of [9], [3], two nontrivial torsion free abelian groups G and H are elementarily equivalent iff for all primes p tf(p, G) = tf(p, H). This gives a natural 1-1 correspondence between the complete theories of nontrivial torsion free abelian groups and the functions /3 : P --~ ~o + 1. /3 will always stand for such a function, and the corresponding complete theory will be denoted by T 0. S o is the set of primes p for which/3(p) ~ 0. For more definitions see section 3. Let 13 be a function from P to o) + 1. We shall study the following questions: Does T o have an elementarily (algebraically) prime model? Does T o have a minimal (strictly minimal) model, and if this is the case, how many such models does T o have? Do the minimal and strictly minimal models of T o coincide? These questions have been studied for the special case of the theory of the

The design of decision procedures for combinations of theories sharing some arithmetic fragment is a challenging problem in verification. One possible solution is to apply a combination method à la Nelson-Oppen, like the one developed by Ghilardi for unions of non-disjoint theories. We show how to apply this non-disjoint combination method with the theory of abelian groups as shared theory. We consider the completeness and the effectiveness of this non-disjoint combination method. For the completeness, we show that the theory of abelian groups can be embedded into a theory admitting quantifier elimination. For achieving effectiveness, we rely on a superposition calculus modulo abelian groups that is shown complete for theories of practical interest in verification.

In the theory of abelian groups, the study of relations between an abelian group and its endomorphism group is an important problem. The question arises on conditions under which an abelian group and its endomorphism group are isomorphic ([1], Problem 31). The answer to this question has been obtained in the classes of completely decomposable abelian groups [2], vector groups [3], in the classes of periodic abelian groups, divisible abelian groups, nonreduced abelian groups, reduced algebraically compact abelian groups [4]. A group A is called an E+-group if it is isomorphic to its endomorphism group End(A). Clearly, if an abelian group A is an E+-group, then its endomorphism group End(A) is also an E+-group. In this paper, we show that the converse statement holds not always. A group A is called an End-group if the following implication takes place:

Major progress in group theory occurred in the nineteenth century but the evolution of group theory began already in the latter part of 18th century. Some characteristic features of 19th century mathematics which had crucial impact on this evolution are concern for rigor and abstraction and the view that mathematics is a human activity without necessarily referring to physical situations. In 1770, Joseph Louis Lagrange (1736-1813) wrote his seminal memoir Reflections on the solution of algebraic equations. He considered ‘abstract’ questions like whether every equation has a root and, if so, how many were real/complex/positive/negative? The problem of algebraically solving 5th degree equations was a major preoccupation (right from the 17th century) to which Lagrange lent his major efforts in this paper. His beautiful idea (now going under the name of Lagrange’s resolvent) is to ‘reduce’ a general equation to auxiliary (resolvent) equations which have one degree less. Later, the theory of finite abelian groups evolved from Carl Friedrich Gauss’s famous “Disquisitiones Arithmeticae”. Gauss (1777-1855) established many of the important properties though he did not use the terminology of group theory. In his work, finite abelian groups4 Math Unlimitedappeared in different forms like the additive group Zn of integers modulo n, the multiplicative group Z∗n of integers modulo n relatively prime to n, the group of equivalence classes of binary quadratic forms, and the group of n-th roots of unity. In 1872, Felix Klein delivered a famous lecture A Comparative Review of Recent Researches in Geometry. The aim of his (so-called) Erlangen Program was the classification of geometry as the study of invariants under various groups of transformations. So, the groups appear here “geometrically” as groups of rigid motions, of similarities, or as the hyperbolic group etc. During the analysis of the connections between the different geometries, the focus was on the study of properties of figures invariant under transformations. Soon after, the focus shifted to a study of the transformations themselves. Thus the study of the geometric relations between figures got converted to the study of the associated transformations. In 1874, Sophus Lie introduced his “theory of continuous transformation groups” what we basically call Lie groups today. Poincaré and Klein began their work on the so-called automorphic functions and the groups associated with them around 1876. Automorphic functions are generalizations of the circular, hyperbolic, elliptic, and other functions of elementary analysis. They are functions of a complex variable z, analytic in some domain, and invariant under the group of transformations x′ = ax+b(a, b, c, d real or complex and ad − bc 0), or under some subgroup of this group. We end this introduction with the ancient problem of finding all those positive integers (called congruent numbers) which are areas of right-angled triangles with rational sides. To this date, the general result describing all of them precisely is not proved. However, the problem can be re-stated in terms of certain groups called elliptic curves and this helps in getting a hold on the problem and in obtaining several partial results. Indeed, if n is a positive integer, then the rational numbers x, y satisfying y2 = x3 − n2x form a group law which can be described geometrically. Then, n is a congruent number if and only if, this group is infinite!