scholarly journals On theory of finite subsets monoid for one torsion abelian group

2021 ◽  
pp. 39-55
Author(s):  
Сергей Михайлович Дудаков
Keyword(s):  
Abelian Group ◽  
Multiplicative Group ◽  
Torsion Group ◽  
Roots Of Unity

Ранее был доказан следующий результат: если абелева группа $\gG$ не является группой кручения, то теория моноида ее конечных подмножеств позволяет интерпретировать элементарную арифметику. В настоящей работе мы приводим пример, который показывает, что аналогичный результат можно получить и, по крайней мере, для некоторых групп кручения. Earlier it was proved the following claim. Let $\gG$ be a non-torsion abelian group and $\gG$ be the semigroup of finite subsets of $\gG$. Then elementary arithmetic can be interpreted in $\gG^*$, so the theory of $\gG^*$ is undecidable. Here we prove the same result for one torsion group, the multiplicative group of all roots of unity.

COMPOSITION SERIES OF THE SOLVABLE ABELIAN FACTOR GROUP SOURCE OF EQUATION OF ALL POLYNOMIAL EQUATIONS

2021 ◽  
Vol 5 (2) ◽  
pp. 462-469
Author(s):  
Bernard Alechenu ◽  
Babayo Muhammed Abdullahi ◽  
Daniel Eneojo Emmanuel
Keyword(s):  
Abelian Group ◽  
Complex Analysis ◽  
Factor Group ◽  
Roots Of Unity ◽  
Composition Series ◽  
Isomorphism Theorem ◽  
Polynomial Equations ◽  
Canonical Map ◽  
The Third ◽  
Recursive Set

This work penciled down the Composition Series of Factor Abelian Group over the source of all polynomial equations gleaned through  the nth roots of unity regular gons on a unit circle, a circle of radius one and centered at zero. To get the composition series, the third isomorphism theorem has to be passed through. But, the third isomorphism theorem itself gleaned via the first which is a deduction of the naturally existing canonical map. The solution of the source atom of the equation of all equation of polynomials are solvable by the intertwine of the Euler’s Formula and the De Moivre’s Theorem which after the inter-math, they become within the domain of complex analysis. For the source root of the equations, there is a recursive set of homomorphisms and ontoness of the mappings geneting the sequential terms in the composition series.    

C-Valuations and Normal C-Orderings

1989 ◽  
Vol 41 (1) ◽  
pp. 14-67 ◽  
Author(s):  
M. Chacron
Keyword(s):  
Abelian Group ◽  
Normal Subgroup ◽  
Division Ring ◽  
Maximal Ideal ◽  
Quotient Group ◽  
Valuation Ring ◽  
Multiplicative Group ◽  
Invertible Elements ◽  
Natural Way

Let D stand for a division ring (or skewfield), let G stand for an ordered abelian group with positive infinity adjoined, and let ω: D → G. We call to a valuation of D with value group G, if ω is an onto mapping from D to G such that(i) ω(x) = ∞ if and only if x = 0,(ii) ω(x1 + x2) = min(ω (x1), ω (x2)), and(iii) ω (x1 x2) = ω (x1) + ω (x2).Associated to the valuation ω are its valuation ringR = ﹛x ∈ Dω(x) ≧ 0﹜,its maximal idealJ = ﹛x ∈ |ω(x) > 0﹜, and its residue division ring D = R/J.The invertible elements of the ring R are called valuation units. Clearly R and, hence, J are preserved under conjugation so that 1 + J is also preserved under conjugation. The latter is thus a normal subgroup of the multiplicative group Dm of D and hence, the quotient group D˙/1 + J makes sense (the residue group of ω). It enlarges in a natural way the residue division ring D (0 excluded, and addition “forgotten“).

Grothendieck groups of quadratic forms and G-equivalence of fields

1973 ◽  
Vol 73 (1) ◽  
pp. 29-36
Author(s):  
K. Szymiczek
Keyword(s):  
Abelian Group ◽  
Quadratic Forms ◽  
Multiplicative Group ◽  
Grothendieck Group ◽  
Grothendieck Groups

Let k be a field of characteristic other than 2 and let g(k) denote the multiplicative group k* of the field k modulo squares, i.e. g(k) = k*/k*2. This is an abelian group of exponent 2 and its order, if finite, is a power of 2. We denote by G(k) the Grothendieck group of quadratic forms over k.

scholarly journals Roots of Elliptic Scator Numbers

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 321
Author(s):  
Manuel Fernandez-Guasti
Keyword(s):  
Abelian Group ◽  
Geometric Interpretation ◽  
Roots Of Unity ◽  
Complex Roots ◽  
De Moivre’S Formula

The Victoria equation, a generalization of De Moivre’s formula in 1+n dimensional scator algebra, is inverted to obtain the roots of a scator. For the qth root in S1+n of a real or a scator number, there are qn possible roots. For n=1, the usual q complex roots are obtained with their concomitant cyclotomic geometric interpretation. For n≥2, in addition to the previous roots, new families arise. These roots are grouped according to two criteria: sets satisfying Abelian group properties under multiplication and sets catalogued according to director conjugation. The geometric interpretation is illustrated with the roots of unity in S1+2.

The Group of Units of the Integral Group Ring ZS3

1972 ◽  
Vol 15 (4) ◽  
pp. 529-534 ◽  
Author(s):  
I. Hughes ◽  
K. R. Pearson
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Group Ring ◽  
Torsion Group ◽  
Integral Group Ring ◽  
Integral Group ◽  
List Type ◽  
Type Number ◽  
Quaternion Group ◽  
Group Of Units

We denote by ZG the integral group ring of the finite group G. We call ±g, for g in G, a trivial unit of ZG. For G abelian, Higman [4] (see also [3, p. 262 ff]) showed that every unit of finite order in ZG is trivial. For arbitrary finite G (indeed, for a torsion group G, not necessarily finite), Higman [4] showed that every unit in ZG is trivial if and only if G is(i) abelian and the order of each element divides 4, or(ii) abelian and the order of each element divides 6, or(iii) the direct product of the quaternion group of order 8 and an abelian group of exponent 2.

On Invariant Means which are Not Inverse Invariant

1968 ◽  
Vol 20 ◽  
pp. 222-224
Author(s):  
M. Rajagopalan ◽  
K. G. Witz
Keyword(s):  
Abelian Group ◽  
Finite Abelian Group ◽  
Torsion Group ◽  
Invariant Mean ◽  
Invariant Means ◽  
Arbitrary Abelian Group

In (1) R. G. Douglas says: “For a finite abelian group there exists a unique invariant mean which must be inversion invariant. For an infinite torsion abelian group it is not clear what the situation is.” It is not hard to see that if every element of an abelian group G is of order 2, then every invariant mean on G is also inversion invariant (see 1, remark 4). In this note we prove the following theorem (Theorem 1 below): An abelian torsion group G has an invariant mean which is not inverse invariant if, and only if, 2G is infinite. This result, together with the theorems of Douglas, answers completely the question of the existence (on an arbitrary abelian group) of invariant means which are not inverse invariant.

The Local Class Group of a Krull Domain

1983 ◽  
Vol 26 (1) ◽  
pp. 13-19 ◽  
Author(s):  
A. Bouvier
Keyword(s):  
Abelian Group ◽  
Quotient Group ◽  
Class Group ◽  
Torsion Group ◽  
Krull Domain ◽  
Krull Domains

AbstractThe local class group of a Krull domain A is the quotient group G(A) = CI(A)/Pic(A). A Krull domain A is locally factorial if and only if G(A) = 0. In this paper, we characterize the Krull domains for which G(A) is a torsion group. We evaluate the local class group of several examples and finally, we explain why every abelian group is the local class group of a Krull domain.

Lipschitz classes on 0-dimensional groups

1967 ◽  
Vol 63 (4) ◽  
pp. 923-928 ◽  
Author(s):  
P. L. Walker
Keyword(s):  
Fourier Transform ◽  
Abelian Group ◽  
Torsion Group ◽  
Character Group ◽  
Lipschitz Classes ◽  
The Fourier Transform

1. Let G be a compact metric 0-dimensional Abelian group. Its dual or character group Γ is a discrete countable torsion group. We denote elements of G by x, of Γ by y, the value of the character y at x by (x, y), and the Fourier transform of f by

scholarly journals On the radical of a group algebra over a commutative ring

1977 ◽  
Vol 18 (1) ◽  
pp. 101-104 ◽  
Author(s):  
Gloria Potter
Keyword(s):  
Abelian Group ◽  
Commutative Ring ◽  
Group Algebra ◽  
Jacobson Radical ◽  
Multiplicative Group ◽  
Commutative Rings ◽  
The Other ◽  
Group Algebras ◽  
Study Group ◽  
The Right

Several people, including Wallace [4] and Passman [3], have studied the Jacobson radical of the group algebra F[G] where F is a field and G is a multiplicative group. In [4], for instance, Wallace proves that if G is an abelian group with Sylow p-subgroup P and if F is a field of characteristic p, then the Jacobson radical of F[G] equals the right ideal generated by the radical of F[P]. In this paper we shall study group algebras over arbitrary commutative rings. By a reduction to the case of a semi-simple commutative ring, we obtain Theorem 1 whose corollary contains a generalization of Wallace's theorem. Theorem 2, on the other hand, uses the first theorem to obtain results related to the main theorem of [3].

scholarly journals On the -theory of -algebras arising from integral dynamics

2016 ◽  
Vol 38 (3) ◽  
pp. 832-862
Author(s):  
SELÇUK BARLAK ◽  
TRON OMLAND ◽  
NICOLAI STAMMEIER
Keyword(s):  
Abelian Group ◽  
Free Abelian Group ◽  
Torsion Group ◽  
Prime Numbers ◽  
Complete Classification ◽  
Greatest Common Divisor ◽  
Direct Sum ◽  
Free Part ◽  
Torsion Part

We investigate the$K$-theory of unital UCT Kirchberg algebras${\mathcal{Q}}_{S}$arising from families$S$of relatively prime numbers. It is shown that$K_{\ast }({\mathcal{Q}}_{S})$is the direct sum of a free abelian group and a torsion group, each of which is realized by another distinct$C^{\ast }$-algebra naturally associated to$S$. The$C^{\ast }$-algebra representing the torsion part is identified with a natural subalgebra${\mathcal{A}}_{S}$of${\mathcal{Q}}_{S}$. For the$K$-theory of${\mathcal{Q}}_{S}$, the cardinality of$S$determines the free part and is also relevant for the torsion part, for which the greatest common divisor$g_{S}$of$\{p-1:p\in S\}$plays a central role as well. In the case where$|S|\leq 2$or$g_{S}=1$we obtain a complete classification for${\mathcal{Q}}_{S}$. Our results support the conjecture that${\mathcal{A}}_{S}$coincides with$\otimes _{p\in S}{\mathcal{O}}_{p}$. This would lead to a complete classification of${\mathcal{Q}}_{S}$, and is related to a conjecture about$k$-graphs.

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