Note on the Definition of an Abelian Group by Independent Postulates

1907 ◽  
Vol 8 (2) ◽  
pp. 94 ◽  
Author(s):  
W. A. Hurwitz
Keyword(s):  
Abelian Group ◽  
Definition Of

scholarly journals A lattice-theoretic characterization of pure subgroups of Abelian groups

2021 ◽  
Author(s):  
M. Ferrara ◽  
M. Trombetti
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Subgroup Lattice ◽  
General Definition ◽  
Short Paper ◽  
Theoretic Characterization ◽  
Definition Of

AbstractLet G be an abelian group. The aim of this short paper is to describe a way to identify pure subgroups H of G by looking only at how the subgroup lattice $$\mathcal {L}(H)$$ L ( H ) embeds in $$\mathcal {L}(G)$$ L ( G ) . It is worth noticing that all results are carried out in a local nilpotent context for a general definition of purity.

scholarly journals A Generalization of Final Rank of Primary Abelian Groups

1970 ◽  
Vol 22 (6) ◽  
pp. 1118-1122 ◽  
Author(s):  
Doyle O. Cutler ◽  
Paul F. Dubois
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Basic Subgroup ◽  
Primary Abelian Group ◽  
Limit Ordinal ◽  
Final Rank ◽  
Definition Of ◽  
Group Recall

Let G be a p-primary Abelian group. Recall that the final rank of G is infn∈ω{r(pnG)}, where r(pnG) is the rank of pnG and ω is the first limit ordinal. Alternately, if Γ is the set of all basic subgroups of G, we may define the final rank of G by supB∈Γ {r(G/B)}. In fact, it is known that there exists a basic subgroup B of G such that r(G/B) is equal to the final rank of G. Since the final rank of G is equal to the final rank of a high subgroup of G plus the rank of pωG, one could obtain the same information if the definition of final rank were restricted to the class of p-primary Abelian groups of length ω.

On a special single-power cyclic hypergroup and its automorphisms

2016 ◽  
Vol 08 (04) ◽  
pp. 1650059 ◽  
Author(s):  
M. Al Tahan ◽  
B. Davvaz
Keyword(s):  
Abelian Group ◽  
Braid Group ◽  
Definition Of ◽  
Single Power

After introducing the definition of hypergroups by Marty, the study of hyperstructures and its applications has been of great importance. In this paper, we find a link between hyperstructures and the infinite non-abelian group, braid group [Formula: see text]. This is the first connection to be done between these two different domains. First, we define a new hyperoperation ⋆ associated to [Formula: see text] and study its properties. Next, we prove that [Formula: see text] is a single-power cyclic hypergroup with infinite period. Then, we define an onto homomorphism from [Formula: see text] to another hypergroup. Finally, we determine the set of all automorphisms of [Formula: see text] and prove that it is a group under the operation of functions composition.

scholarly journals Local polynomial functions on factor rings of the integers

1979 ◽  
Vol 27 (2) ◽  
pp. 232-238 ◽  
Author(s):  
Hans Lausch ◽  
Wilfried Nöbauer
Keyword(s):  
Abelian Group ◽  
Universal Algebra ◽  
Polynomial Function ◽  
Commutative Rings ◽  
Infinite Length ◽  
Polynomial Functions ◽  
Local Polynomial ◽  
Definition Of ◽  
Image Position

AbstractLet A be a universal algebra. A function ϕ Ak-A is called a t-local polynomial function, if ϕ can ve interpolated on any t places of Ak by a polynomial function— for the definition of a polynomial function on A, see Lausch and Nöbauer (1973), Let Pk(A) be the set of the polynomial functions, LkPk(A) the set of all t-local polynmial functions on A and LPk(A) the intersection of all LtPk(A), then . If A is an abelian group, then this chain has at most five distinct members— see Hule and Nöbauer (1977)— and if A is a lattice, then it has at most three distinct members— see Dorninger and Nöbauer (1978). In this paper we show that in the case of commutative rings with identity there does not exist such a bound on the length of the chain and that, in this case, there exist chains of even infinite length.

scholarly journals On the dimension of an Abelian group

2021 ◽  
Vol 27 (4) ◽  
pp. 267-275
Author(s):  
Timo Tossavainen ◽  
◽  
Pentti Haukkanen ◽  
Keyword(s):  
Abelian Group ◽  
Dimensional Structure ◽  
Torsion Subgroup ◽  
Finitely Generated ◽  
Definition Of

We introduce a measure of dimensionality of an Abelian group. Our definition of dimension is based on studying perpendicularity relations in an Abelian group. For G ≅ ℤn, dimension and rank coincide but in general they are different. For example, we show that dimension is sensitive to the overall dimensional structure of a finite or finitely generated Abelian group, whereas rank ignores the torsion subgroup completely.

Trace in additive categories

1970 ◽  
Vol 67 (3) ◽  
pp. 541-547 ◽  
Author(s):  
Alan Thomas
Keyword(s):  
Abelian Group ◽  
Commutative Diagram ◽  
Additive Category ◽  
Definition Of ◽  
Image Position

1. Let ℳ be an additive category. (We refer to ((1), Ch. IX) for the definition of an additive category and associated terms. In particular a sequenceis exact if i = kerp and p = coker i.) We write End M for Hom(M, M). A trace on ℳ with values in an abelian group G is a collection of (abelian group) homomorphismsone for each M ∈ ℳ, satisfying the following two conditions:(i) Exactness. Given a commutative diagram with exact rows,then tA(f) + tC(h) = tB(g).

The distribution of sequences in Abelian groups

1970 ◽  
Vol 67 (1) ◽  
pp. 1-11 ◽  
Author(s):  
J. S. Dennis
Keyword(s):  
Abelian Group ◽  
Uniform Distribution ◽  
Abelian Groups ◽  
Definition Of ◽  
Independent Distribution ◽  
Prime Exponent

1Introduction. In this paper, I give a definition of uniform distribution for sequences taking values in certain types of Abelian group—in particular, in groups of prime exponent—and one of independent distribution for sets of such sequences. In the various cases, I find conditions for the properties to hold, which are similar to Weyl's criterion for the uniform distribution of real sequences modulo 1.

scholarly journals q-Binomial Convolution and Transformations of q-Appell Polynomials

Axioms ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 70
Author(s):  
Alaa Mohammed Obad ◽  
Asif Khan ◽  
Kottakkaran Sooppy Nisar ◽  
Ahmed Morsy
Keyword(s):  
Abelian Group ◽  
Group Structure ◽  
Random Variable ◽  
Scale Transformation ◽  
Appell Polynomials ◽  
Quantum Calculus ◽  
Appell Sequences ◽  
Binomial Convolution ◽  
Definition Of ◽  
Determinantal Form

In this paper, binomial convolution in the frame of quantum calculus is studied for the set Aq of q-Appell sequences. It has been shown that the set Aq of q-Appell sequences forms an Abelian group under the operation of binomial convolution. Several properties for this Abelian group structure Aq have been studied. A new definition of the q-Appell polynomials associated with a random variable is proposed. Scale transformation as well as transformation based on expectation with respect to a random variable is used to present the determinantal form of q-Appell sequences.

scholarly journals Cohomological invariants of representations of 3-manifold groups

2020 ◽  
pp. 2043003
Author(s):  
Haimiao Chen
Keyword(s):  
Abelian Group ◽  
Discrete Group ◽  
Representation Formula ◽  
Practical Method ◽  
Cohomological Invariants ◽  
Continuous Map ◽  
Definition Of ◽  
Manifolds With Corners ◽  
Chern Simons

Suppose [Formula: see text] is a discrete group, and [Formula: see text], with [Formula: see text] an abelian group. Given a representation [Formula: see text], with [Formula: see text] a closed 3-manifold, put [Formula: see text], where [Formula: see text] is a continuous map inducing [Formula: see text] which is unique up to homotopy, and [Formula: see text] is the pairing. We extend the definition of [Formula: see text] to manifolds with corners, and establish a gluing law. Based on these, we present a practical method for computing [Formula: see text] when [Formula: see text] is given by a surgery along a link [Formula: see text]. In particular, the Chern–Simons invariant can be computed this way.

Galois Extensions as Modules over the Group Ring

1970 ◽  
Vol 22 (2) ◽  
pp. 242-248 ◽  
Author(s):  
Gerald Garfinkel ◽  
Morris Orzech
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Galois Group ◽  
Commutative Ring ◽  
Group Ring ◽  
Finite Abelian Group ◽  
Galois Extensions ◽  
Normal Bases ◽  
Direct Sum ◽  
Definition Of

Suppose that R is a commutative ring and G is a finite abelian group. In § 2 we review the definition of E(R, G) (T(R, G)), the group of all (commutative) Galois extensions S of R with Galois group G. We discuss the properties of these groups as functors of G and give an example which exhibits some of the pathological properties of the functor E(R, – ). In § 3 we display a homomorphism from E(R, G) to Pic (R(G)); we use this homomorphism to prove that if S is commutative, G has exponent m, and R(G) has Serre dimension 0 or 1, then a direct sum of m copies of S is isomorphic as a G-module to a direct sum of m copies of R(G). (This result is related to [5, Theorem 4.2], where it is shown that if S is a free R-module and G is any finite group with n elements, then Sn is isomorphic to R(G)n as G-modules.) We also give some examples of Galois extensions without normal bases.

Sign in / Sign up

Export Citation Format

Share Document