Умножения на смешанных абелевых группах

The splitting length of a mixed Abelian group G is defined as the smallest positive integer n such that \( \mathop \otimes \limits^n G \) splits. The task of determining the splitting length of mixed Abelian groups was formulated by Irwin, Khabbaz, and Rayna. In this paper, a criterion for determining whether \( \mathop \otimes \limits^n G \) splits for countable mixed Abelian groups G of torsion-free rank 1 is found.

Two operators or two subgroups of a group will be called I-conjugate whenever they are conjugate under the group of isomorphisms of this group. It is well known that all the cyclic subgroups of highest order contained in any abelian group are I-conjugate. A necessary and sufficient condition that all the cyclic subgroups of any other order contained in an abelian group G of order pmm, p being a prime number, are I-conjugate is that all the invariants of G are equal to each other. When this condition is satisfied we shall prove, in particular, that the number of the subgroups of order pa, a c m , contained in G is equal to the number of its subgroups of index pa. It is known that the ?-subgroup of G is composed of the pth power of every operator of G. The concept of +-subgroup in connection with abelian groups can readily be extended by considering the characteristic subgroup of G composed of the path power of every operator of G. This subgroup may be called the Oa-subgroup of G. It is composed of all the operators of G which are found in each one of its subgroups of index pa, and all the operators of highest order in a 4.-subgroup of G are I-conjugate. In particular, the 0&-subgroup generated by operators of order p is the fundamental characteristic subgroup of G. t In fact, when p > 2 the 0a-subgroup which is generated by operators of order pr is contained in every characteristic subgroup of G which involves operators of order pr. The quotient group of G corresponding to a 4a-subgroup may be called a 0,fi-quotient group. This quotient group is simply isomorphic with the subgroup of G generated bv all its operators whose orders divide p> Hence the following: THEOREM. The number of the subgroups of index pa contained in any abelian

Let $S$ be a subsemigroup which contains 0 of a torsion-free abelian (additive) group. Then $S$ is called a grading monoid (or a $g$-monoid). The group $ \{s-s'|s,s'\in S\}$ is called the quotient group of $S$, and is denored by $q(S)$. Let $R$ be a commutative ring. The total quotient ring of $R$ is denoted by $q(R)$. Throught the paper, we assume that a $g$-monoid properly contains $ \{0\}$. A commutative ring is called a ring, and a non-zero-divisor of a ring is called a regular element of the ring. We consider integral elements over the semigroup ring $ R[X;S]$ of $S$ over $R$. Let $S$ be a $g$-monoid with quotient group $G$. If $ n\alpha\in S$ for an element $ \alpha$ of $G$ and a natural number $n$ implies $ \alpha\in S$, then $S$ is called an integrally closed semigroup. We know the following fact: ${\bf Theorem~1}$ ([G2, Corollary 12.11]). Let $D$ be an integral domain and $S$ a $g$-monoid. Then $D[X;S]$ is integrally closed if and only if $D$ is an integrally closed domain and $S$ is an integrally closed semigroup. Let $R$ be a ring. In this paper, we show that conditions for $R[X;S]$ to be integrally closed reduce to conditions for the polynomial ring of an indeterminate over a reduced total quotient ring to be integrally closed (Theorem 15). Clearly the quotient field of an integral domain is a von Neumann regular ring. Assume that $q(R)$ is a von Neumann regular ring. We show that $R[X;S]$ is integrally closed if and only if $R$ is integrally closed and $S$ is integrally closed (Theorem 20). Let $G$ be a $g$-monoid which is a group. If $R$ is a subring of the ring $T$ which is integrally closed in $T$, we show that $R[X;G]$ is integrally closed in $T[X;S]$ (Theorem 13). Finally, let $S$ be sub-$g$-monoid of a totally ordered abelian group. Let $R$ be a subring of the ring $T$ which is integrally closed in $T$. If $g$ and $h$ are elements of $T[X;S]$ with $h$ monic and $gh\in R[X;S]$, we show that $g\in R[X;S]$ (Theorem 24).

Abelian groups is the most famous in group theory and is also one of the most important groups, the study of abelian groups occupies a very important position in the group theory, abelian groups for almost all the research plays a backbone role, this article through replace new notation to the abelian groups of the term for a group, introduce some new definitions such as p-primary group, p-primary components of G, independent of groups and the Sylow subgroups for tool are discussed, and finally got about finite abelian groups of several important conclusions.

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:

In this chapter we develop the algebra underlying multirate filter structures over free abelian groups of finite rank. Suppose A is a free abelian group of rank N. Each subgroup Δ of A is a free abelian group of rank n ≤ N. The basis theorem describes how Δ sits in A in the sense that it characterizes the quotient group A/Δ. This quotient group can be a finite abelian group a free abelian group of finite rank the direct product of a finite abelian group and a free abelian group of finite rank.

(1984). The theory of separable mixed abelian groups. Communications in Algebra: Vol. 12, No. 15, pp. 1813-1834.

Supercharacter theories of semiextraspecial p-groups and Frobenius 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.

In this chapter, we discuss several special types of modules that will make frequent appearances later. These include semisimple modules, artinian modules, and torsionfree modules, which have prototypes in vector spaces, finite abelian groups, and torsionfree abelian groups, respectively. That special types of modules have useful roles in the study of arbitrary modules may be seen already in the case of abelian groups (i.e., ℤ-modules). In studying an arbitrary abelian group, an almost reflexive first step is to look at its torsion part (i.e., the torsion subgroup) and its torsionfree part (i.e., the factor group modulo the torsion subgroup), since entirely different techniques are available (and needed) for dealing with torsion groups and torsionfree groups. On the torsionfree side, vector spaces make an appearance due to the facts that the torsionfree divisible abelian groups are exactly the vector spaces over ℚ and that every torsionfree abelian group can be embedded in a divisible one. On the torsion side, many questions can be reduced to the case of finite abelian groups, since every torsion abelian group is a directed union of finite subgroups. In studying torsion abelian groups, one also reduces to the case of p -groups for various primes p . Vector spaces make another appearance here, since in an abelian p -group the set of elements of order p (together with 0) forms a vector space over ℤ/ p ℤ.

On the computational complexity of the theory of Abelian groups

Brouéʼs abelian defect group conjecture holds for the sporadic simple Conway group [formula omitted

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.

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

We show that the cohomology ring of a finite-dimensional complex pointed Hopf algebra with an abelian group of group-like elements is finitely generated. Our strategy has three major steps. We first reduce the problem to the finite generation of cohomology of finite dimensional Nichols algebras of diagonal type. For the Nichols algebras, we do a detailed analysis of cohomology via the Anick resolution reducing the problem further to specific combinatorial properties. Finally, to check these properties, we turn to the classification of Nichols algebras of diagonal type due to Heckenberger. In this paper, we complete the verification of these combinatorial properties for major parametric families, including Nichols algebras of Cartan and super types and develop all the theoretical foundations necessary for the case-by-case analysis. The remaining discrete families are addressed in a separate publication. As an application of the main theorem, we deduce finite generation of cohomology for other classes of finite-dimensional Hopf algebras, including basic Hopf algebras with abelian groups of characters and finite quotients of quantum groups at roots of one.