scholarly journals Every finite abelian group is a subgroup of the additive group of a finite simple left brace

2021 ◽  
Vol 225 (1) ◽  
pp. 106476
Author(s):  
F. Cedó ◽  
E. Jespers ◽  
J. Okniński
Keyword(s):  
Abelian Group ◽  
Finite Abelian Group ◽  
Additive Group

scholarly journals Classification of Finite Rings of Orderp6by Generators and Relations

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
P. Karimi Beiranvand ◽  
R. Beyranvand ◽  
M. Gholami
Keyword(s):  
Abelian Group ◽  
Prime Number ◽  
Binary Operation ◽  
Finite Abelian Group ◽  
Sufficient Conditions ◽  
Additive Group ◽  
Finite Rings ◽  
Generators And Relations ◽  
Necessary And Sufficient

For any finite abelian group(R,+), we define a binary operation or “multiplication” onRand give necessary and sufficient conditions on this multiplication forRto extend to a ring. Then we show when two rings made on the same group are isomorphic. In particular, it is shown that there aren+1rings of orderpnwith characteristicpn, wherepis a prime number. Also, all finite rings of orderp6are described by generators and relations. Finally, we give an algorithm for the computation of all finite rings based on their additive group.

Additive bases via Fourier analysis

2021 ◽  
pp. 1-12
Author(s):  
Bodan Arsovski
Keyword(s):  
Abelian Group ◽  
Fourier Analysis ◽  
Vector Space ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Probabilistic Method ◽  
Additive Basis ◽  
Generating Sets

Abstract Extending a result by Alon, Linial, and Meshulam to abelian groups, we prove that if G is a finite abelian group of exponent m and S is a sequence of elements of G such that any subsequence of S consisting of at least $$|S| - m\ln |G|$$ elements generates G, then S is an additive basis of G . We also prove that the additive span of any l generating sets of G contains a coset of a subgroup of size at least $$|G{|^{1 - c{ \in ^l}}}$$ for certain c=c(m) and $$ \in = \in (m) < 1$$ ; we use the probabilistic method to give sharper values of c(m) and $$ \in (m)$$ in the case when G is a vector space; and we give new proofs of related known results.

Representation of zero-sum invariants by sets of zero-sum sequences over a finite abelian group

2021 ◽  
Author(s):  
Weidong Gao ◽  
Siao Hong ◽  
Wanzhen Hui ◽  
Xue Li ◽  
Qiuyu Yin ◽  
...  
Keyword(s):  
Abelian Group ◽  
Finite Abelian Group ◽  
Zero Sum

Stable cohomotopy and cobordism of abelian groups

1981 ◽  
Vol 90 (2) ◽  
pp. 273-278 ◽  
Author(s):  
C. T. Stretch
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Formal Group ◽  
Characteristic Class ◽  
Burnside Ring ◽  
Classifying Space ◽  
Formal Group Law ◽  
Stable Cohomotopy ◽  
Group Law

The object of this paper is to prove that for a finite abelian group G the natural map is injective, where Â(G) is the completion of the Burnside ring of G and σ0(BG) is the stable cohomotopy of the classifying space BG of G. The map â is detected by means of an M U* exponential characteristic class for permutation representations constructed in (11). The result is a generalization of a theorem of Laitinen (4) which treats elementary abelian groups using ordinary cohomology. One interesting feature of the present proof is that it makes explicit use of the universality of the formal group law of M U*. It also involves a computation of M U*(BG) in terms of the formal group law. This may be of independent interest. Since writing the paper the author has discovered that M U*(BG) has previously been calculated by Land-weber(5).

CANCELLATION LAW AND UNIQUE FACTORIZATION THEOREM FOR STRING OPERATIONS

2006 ◽  
Vol 05 (02) ◽  
pp. 231-243
Author(s):  
DONGVU TONIEN
Keyword(s):  
Abelian Group ◽  
Algebraic Structure ◽  
Additive Group ◽  
New Product ◽  
Binary String ◽  
Factorization Theorem ◽  
General Criterion ◽  
Unique Factorization ◽  
Product Operation ◽  
Associative Law

Recently, Hoit introduced arithmetic on blocks, which extends the binary string operation by Jacobs and Keane. A string of elements from the Abelian additive group of residues modulo m, (Zm, ⊕), is called an m-block. The set of m-blocks together with Hoit's new product operation form an interesting algebraic structure where associative law and cancellation law hold. A weaker form of unique factorization and criteria for two indecomposable blocks to commute are also proved. In this paper, we extend Hoit's results by replacing the Abelian group (Zm, ⊕) by an arbitrary monoid (A, ◦). The set of strings built up from the alphabet A is denoted by String(A). We extend the operation ◦ on the alphabet set A to the string set String(A). We show that (String(A), ◦) is a monoid if and only if (A, ◦) is a monoid. When (A, ◦) is a group, we prove that stronger versions of a cancellation law and unique factorization hold for (String(A), ◦). A general criterion for two irreducible strings to commute is also presented.

An improved query for the hidden subbroup problem

2014 ◽  
Vol 14 (5&6) ◽  
pp. 467-492
Author(s):  
Asif Shakeel
Keyword(s):  
Abelian Group ◽  
Standard Method ◽  
Finite Abelian Group ◽  
Quantum Algorithms ◽  
Success Probability ◽  
Hidden Subgroup Problem ◽  
Subgroup Identification ◽  
Subgroup Problem ◽  
Query Algorithms

The Hidden Subgroup Problem (HSP) is at the forefront of problems in quantum algorithms. In this paper, we introduce a new query, the \textit{character} query, generalizing the well-known phase kickback trick that was first used successfully to efficiently solve Deutsch's problem. An equal superposition query with $\vert 0 \rangle$ in the response register is typically used in the ``standard method" of single-query algorithms for the HSP. The proposed character query improves over this query by maximizing the success probability of subgroup identification under a uniform prior, for the HSP in which the oracle functions take values in a finite abelian group. We apply our results to the case when the subgroups are drawn from a set of conjugate subgroups and obtain a success probability greater than that found by Moore and Russell.

scholarly journals On Subsequence Sums of a Zero-sum Free Sequence

10.37236/970 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Fang Sun
Keyword(s):  
Abelian Group ◽  
Finite Abelian Group ◽  
Zero Sum ◽  
Open Question

Let $G$ be a finite abelian group with exponent $m$, and let $S$ be a sequence of elements in $G$. Let $f(S)$ denote the number of elements in $G$ which can be expressed as the sum over a nonempty subsequence of $S$. In this paper, we show that, if $|S|=m$ and $S$ contains no nonempty subsequence with zero sum, then $f(S)\geq 2m-1$. This answers an open question formulated by Gao and Leader. They proved the same result with the restriction $(m,6)=1$.

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