scholarly journals DETECTING STEINER AND LINEAR ISOMETRIES OPERADS

2020 ◽  
pp. 1-36
Author(s):  
JONATHAN RUBIN
Keyword(s):  
Abelian Group ◽  
Partial Order ◽  
Cyclic Group ◽  
Prime Power ◽  
Finite Abelian Group ◽  
Finite Cyclic Group ◽  
Indexing System ◽  
Linear Isometries

Abstract We study the indexing systems that correspond to equivariant Steiner and linear isometries operads. When G is a finite abelian group, we prove that a G-indexing system is realized by a Steiner operad if and only if it is generated by cyclic G-orbits. When G is a finite cyclic group, whose order is either a prime power or a product of two distinct primes greater than 3, we prove that a G-indexing system is realized by a linear isometries operad if and only if it satisfies Blumberg and Hill’s horn-filling condition. We also repackage the data in an indexing system as a certain kind of partial order. We call these posets transfer systems, and develop basic tools for computing with them.

On (k, l)-sets in cyclic groups of odd prime order

2001 ◽  
Vol 63 (1) ◽  
pp. 115-121 ◽  
Author(s):  
T. Bier ◽  
A. Y. M. Chin
Keyword(s):  
Abelian Group ◽  
Cyclic Group ◽  
Arithmetic Progression ◽  
Prime Order ◽  
Finite Abelian Group ◽  
Maximum Cardinality ◽  
Cyclic Groups ◽  
Positive Integers

Let A be a finite Abelian group written additively. For two positive integers k, l with k ≠ l, we say that a subset S ⊂ A is of type (k, l) or is a (k, l) -set if the equation x1 + x2 + … + xk − xk+1−… − xk+1 = 0 has no solution in the set S. In this paper we determine the largest possible cardinality of a (k, l)-set of the cyclic group ℤP where p is an odd prime. We also determine the number of (k, l)-sets of ℤp which are in arithmetic progression and have maximum cardinality.

scholarly journals Subsums of a Zero-sum Free Subset of an Abelian Group

10.37236/840 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Weidong Gao ◽  
Yuanlin Li ◽  
Jiangtao Peng ◽  
Fang Sun
Keyword(s):  
Abelian Group ◽  
Lower Bound ◽  
Cyclic Group ◽  
Open Problem ◽  
Nonempty Subset ◽  
Finite Abelian Group ◽  
Positive Answer ◽  
Free Subset ◽  
Zero Sum ◽  
G 28

Let $G$ be an additive finite abelian group and $S \subset G$ a subset. Let f$(S)$ denote the number of nonzero group elements which can be expressed as a sum of a nonempty subset of $S$. It is proved that if $|S|=6$ and there are no subsets of $S$ with sum zero, then f$(S)\geq 19$. Obviously, this lower bound is best possible, and thus this result gives a positive answer to an open problem proposed by R.B. Eggleton and P. Erdős in 1972. As a consequence, we prove that any zero-sum free sequence $S$ over a cyclic group $G$ of length $|S| \ge {6|G|+28\over19}$ contains some element with multiplicity at least ${6|S|-|G|+1\over17}$.

On ∗-clean group rings over abelian groups

2016 ◽  
Vol 16 (08) ◽  
pp. 1750152 ◽  
Author(s):  
Dongchun Han ◽  
Yuan Ren ◽  
Hanbin Zhang
Keyword(s):  
Cyclic Group ◽  
Prime Power ◽  
Associative Ring ◽  
Finite Abelian Group ◽  
Sufficient Conditions ◽  
Complete Characterization ◽  
Necessary Condition ◽  
Group Rings ◽  
Prime Power Order

An associative ring with unity is called clean if each of its elements is the sum of an idempotent and a unit. A clean ring with involution ∗ is called ∗-clean if each of its elements is the sum of a unit and a projection (∗-invariant idempotent). In a recent paper, Huang, Li and Yuan provided a complete characterization that when a group ring [Formula: see text] is ∗-clean, where [Formula: see text] is a finite field and [Formula: see text] is a cyclic group of an odd prime power order [Formula: see text]. They also provided a necessary condition and a few sufficient conditions for [Formula: see text] to be ∗-clean, where [Formula: see text] is a cyclic group of order [Formula: see text]. In this paper, we extend the above result of Huang, Li and Yuan from [Formula: see text] to [Formula: see text] and provide a characterization of ∗-clean group rings [Formula: see text], where [Formula: see text] is a finite abelian group and [Formula: see text] is a field with characteristic not dividing the exponent of [Formula: see text].

scholarly journals INTERCHANGE RINGS

2016 ◽  
Vol 101 (3) ◽  
pp. 310-334
Author(s):  
CHARLES C. EDMUNDS
Keyword(s):  
Abelian Group ◽  
Prime Power ◽  
Binary Operation ◽  
Finite Abelian Group ◽  
Linear Operators ◽  
Prime Power Order ◽  
Cyclic Groups ◽  
Isomorphism Classes ◽  
Direct Sum ◽  
Interchange Law

An interchange ring,$(R,+,\bullet )$, is an abelian group with a second binary operation defined so that the interchange law$(w+x)\bullet (y+z)=(w\bullet y)+(x\bullet z)$ holds. An interchange near ring is the same structure based on a group which may not be abelian. It is shown that each interchange (near) ring based on a group $G$ is formed from a pair of endomorphisms of $G$ whose images commute, and that all interchange (near) rings based on $G$ can be characterized in this manner. To obtain an associative interchange ring, the endomorphisms must be commuting idempotents in the endomorphism semigroup of $G$. For $G$ a finite abelian group, we develop a group-theoretic analogue of the simultaneous diagonalization of idempotent linear operators and show that pairs of endomorphisms which yield associative interchange rings can be diagonalized and then put into a canonical form. A best possible upper bound of $4^{r}$ can be given for the number of distinct isomorphism classes of associative interchange rings based on a finite abelian group $A$ which is a direct sum of $r$ cyclic groups of prime power order. If $A$ is a direct sum of $r$ copies of the same cyclic group of prime power order, we show that there are exactly ${\textstyle \frac{1}{6}}(r+1)(r+2)(r+3)$ distinct isomorphism classes of associative interchange rings based on $A$. Several examples are given and further comments are made about the general theory of interchange rings.

scholarly journals ON THE MAXIMUM SIZE OF A (k,l)-SUM-FREE SUBSET OF AN ABELIAN GROUP

2009 ◽  
Vol 05 (06) ◽  
pp. 953-971 ◽  
Author(s):  
BÉLA BAJNOK
Keyword(s):  
Abelian Group ◽  
Cyclic Group ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Maximum Size ◽  
Finite Abelian Groups ◽  
Free Subset ◽  
Free Set ◽  
Free Sets

A subset A of a given finite abelian group G is called (k,l)-sum-free if the sum of k (not necessarily distinct) elements of A does not equal the sum of l (not necessarily distinct) elements of A. We are interested in finding the maximum size λk,l(G) of a (k,l)-sum-free subset in G. A (2,1)-sum-free set is simply called a sum-free set. The maximum size of a sum-free set in the cyclic group ℤn was found almost 40 years ago by Diamanda and Yap; the general case for arbitrary finite abelian groups was recently settled by Green and Ruzsa. Here we find the value of λ3,1(ℤn). More generally, a recent paper by Hamidoune and Plagne examines (k,l)-sum-free sets in G when k - l and the order of G are relatively prime; we extend their results to see what happens without this assumption.

CHARACTERIZATIONS OF SOME WREATH PRODUCTS OF GROUPS BY THEIR ENDOMORPHISM SEMIGROUPS

2008 ◽  
Vol 18 (02) ◽  
pp. 243-255 ◽  
Author(s):  
PEETER PUUSEMP
Keyword(s):  
Abelian Group ◽  
Wreath Product ◽  
Cyclic Group ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Wreath Products ◽  
Endomorphism Semigroup ◽  
Finite Abelian Groups ◽  
Products Of Groups

Let A be a cyclic group of order pn, where p is a prime, and B be a finite abelian group or a finite p-group which is determined by its endomorphism semigroup in the class of all groups. It is proved that under these assumptions the wreath product A Wr B is determined by its endomorphism semigroup in the class of all groups. It is deduced from this result that if A, B, A0,…, An are finite abelian groups and A0,…, An are p-groups, p prime, then the wreath products A Wr B and An Wr (…( Wr (A1 Wr A0))…) are determined by their endomorphism semigroups in the class of all groups.

scholarly journals Inverse results for weighted Harborth constants

2016 ◽  
Vol 12 (07) ◽  
pp. 1845-1861 ◽  
Author(s):  
Luz E. Marchan ◽  
Oscar Ordaz ◽  
Dennys Ramos ◽  
Wolfgang A. Schmid
Keyword(s):  
Inverse Problem ◽  
Abelian Group ◽  
Inverse Problems ◽  
Cyclic Group ◽  
Finite Abelian Group ◽  
Maximal Length ◽  
Direct Sum ◽  
Inverse Results

For a finite abelian group [Formula: see text], the Harborth constant is defined as the smallest integer [Formula: see text] such that each squarefree sequence over [Formula: see text] of length [Formula: see text] has a subsequence of length equal to the exponent of [Formula: see text] whose terms sum to [Formula: see text]. The plus-minus weighted Harborth constant is defined in the same way except that the existence of a plus-minus weighted subsum equaling [Formula: see text] is required, that is, when forming the sum one can choose a sign for each term. The inverse problem associated to these constants is the problem of determining the structure of squarefree sequences of maximal length that do not yet have such a zero-subsum. We solve the inverse problems associated to these constants for certain groups, in particular, for groups that are the direct sum of a cyclic group and a group of order two. Moreover, we obtain some results for the plus-minus weighted Erdős–Ginzburg–Ziv constant.

scholarly journals MINIMAL ZERO-SUM SEQUENCES OF LENGTH FOUR OVER FINITE CYCLIC GROUPS II

2013 ◽  
Vol 09 (04) ◽  
pp. 845-866 ◽  
Author(s):  
YUANLIN LI ◽  
JIANGTAO PENG
Keyword(s):  
Recent Result ◽  
Cyclic Group ◽  
Open Problem ◽  
Prime Power ◽  
Prime Numbers ◽  
Affirmative Answer ◽  
Cyclic Groups ◽  
Finite Cyclic Group ◽  
Zero Sum

Let G be a finite cyclic group. Every sequence S over G can be written in the form S = (n1g)⋅…⋅(nlg) where g ∈ G and n1, …, nl ∈ [1, ord (g)], and the index ind (S) of S is defined to be the minimum of (n1+⋯+nl)/ ord (g) over all possible g ∈ G such that 〈g〉 = G. An open problem on the index of length four sequences asks whether or not every minimal zero-sum sequence of length 4 over a finite cyclic group G with gcd (|G|, 6) = 1 has index 1. In this paper, we show that if G = 〈g〉 is a cyclic group with order of a product of two prime powers and gcd (|G|, 6) = 1, then every minimal zero-sum sequence S of the form S = (g)(n2g)(n3g)(n4g) has index 1. In particular, our result confirms that the above problem has an affirmative answer when the order of G is a product of two different prime numbers or a prime power, extending a recent result by the first author, Plyley, Yuan and Zeng.

scholarly journals On finite loops whose inner mapping groups are Abelian II

2005 ◽  
Vol 71 (3) ◽  
pp. 487-492
Author(s):  
Markku Niemenmaa
Keyword(s):  
Abelian Group ◽  
Direct Product ◽  
Prime Power ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Prime Power Order ◽  
Finite Abelian Groups ◽  
Cyclic Groups ◽  
Inner Mapping Group ◽  
Mapping Group

If the inner mapping group of a loop is a finite Abelian group, then the loop is centrally nilpotent. We first investigate the structure of those finite Abelian groups which are not isomorphic to inner mapping groups of loops and after this we show that if the inner mapping group of a loop is isomorphic to the direct product of two cyclic groups of the same odd prime power order pn, then our loop is centrally nilpotent of class at most n + 1.

scholarly journals Modification of Griffiths' Result for Even Integers

10.37236/6285 ◽  
2016 ◽  
Vol 23 (4) ◽  
Author(s):  
Eshita Mazumdar ◽  
Sneh Bala Sinha
Keyword(s):  
Abelian Group ◽  
Cyclic Group ◽  
Finite Abelian Group ◽  
General Bound ◽  
Zero Sum

For a finite abelian group $G$ with $\exp(G)=n$, the arithmetical invariant $\mathsf s_A(G)$ is defined to be the least integer $k$ such that any sequence $S$ with length $k$ of elements in $G$ has a $A$ weighted zero-sum subsequence of length $n$. When $A=\{1\}$, it is the Erdős-Ginzburg-Ziv constant and is denoted by $\mathsf s (G)$. For certain class of sets $A$, we already have some general bounds for these weighted constants corresponding to the cyclic group $\mathbb{Z}_n$, which was given by Griffiths. For odd integer $n$, Adhikari and Mazumdar generalized the above mentioned results in the sense that they hold for more sets $A$. In the present paper we modify Griffiths' method for even $n$ and obtain general bound for the weighted constants for certain class of weighted sets which include sets that were not covered by Griffiths for $n\equiv 0 \pmod{4}$.

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