Finite Abelian Research Articles

On the Cohopficity of the Direct Product of Cohopfian Groups

Suppose both A and B are cohopfian groups. Then A × B is cohopfian if A is either extremely noncommutative and torsion free, or finite Abelian, or finite simple.

Generalized Bhaskar Rao Designs with Block Size 3 over Finite Abelian Groups

We show that if G is a finite Abelian group and the block size is 3, then the necessary conditions for the existence of a (v,3,λ;G) GBRD are sufficient. These necessary conditions include the usual necessary conditions for the existence of the associated (v,3,λ) BIBD plus λ≡ 0 (mod|G|), plus some extra conditions when |G| is even, namely that the number of blocks be divisible by 4 and, if v = 3 and the Sylow 2-subgroup of G is cyclic, then also λ≡ 0 (mod2|G|).

Cayley Digraphs of Finite Abelian Groups and Monomial Ideals

In the study of double-loop computer networks, the diagrams known as L-shapes arise as a graphical representation of an optimal routing for every graph's node. The description of these diagrams provides an efficient method for computing the diameter and the average minimum distance of the corresponding graphs. We extend these diagrams to multiloop computer networks. For each Cayley digraph with a finite abelian group as vertex set, we define a monomial ideal and consider its representations via its minimal system of generators or its irredundant irreducible decomposition. From this last piece of information, we can compute the graph's diameter and average minimum distance. That monomial ideal is the initial ideal of a certain lattice with respect to a graded monomial ordering. This result permits the use of Gröbner bases for computing the ideal and finding an optimal routing. Finally, we present a family of Cayley digraphs parametrized by their diameter d, all of them associated to irreducible monomial ideals.

Finite Abelian subgroups of the Cremona group of the plane

We present in this Note some results on conjugacy classes of finite Abelian subgroups of the Cremona group of the plane. To cite this article: J. Blanc, C. R. Acad. Sci. Paris, Ser. I 344 (2007).

Generic quantum Fourier transforms

The quantum Fourier transform (QFT) is a principal ingredient appearing in many efficient quantum algorithms. We present a generic framework for the construction of efficient quantum circuits for the QFT by “quantizing” the highly successful separation of variables technique for the construction of efficient classical Fourier transforms. Specifically, we apply Bratteli diagrams, Gel'fand-Tsetlin bases, and strong generating sets of small adapted diameter to provide efficient quantum circuits for the QFT over a wide variety of finite Abelian and non-Abelian groups, including all families of groups for which efficient QFTs are currently known and many new families as well. Moreover, our method provides the first subexponential-size quantum circuits for the QFT over the linear groups GL k ( q ), SL k ( q ), and the finite groups of Lie type, for any fixed prime power q .

Varieties of finite supersolvable groups with the M. Hall property

The varieties in the title are shown to be precisely the product varieties Gp*Ab(d) for some prime p and some positive integer d dividing p−1. Here Gp denotes the variety of all finite p-groups and Ab(d) the variety of all finite Abelian groups of exponent dividing d. It turns out that these are exactly those varieties H of supersolvable groups for which all finitely generated free pro-H groups are freely indexed in the sense of Lubotzky and van den Dries. Several alternative characterizations of these varieties are presented. Some applications to formal language theory and finite monoid theory are also given. Among these is the determination of all supersolvable solutions H to the equations PH = J*H and J*H = JOpen image in new windowH which is, to the present date, the most complete solution to a problem raised by Pin. Another consequence of our results is that for each such variety H the monoid variety PH = J*H = JOpen image in new windowH has decidable membership.

Full Rank Tilings of Finite Abelian Groups

A tiling of a finite abelian group $G$ is a pair $(V,A)$ of subsets of $G$ such that $0$ is in both $V$ and $A$ and every $g\in G$ can be uniquely written as $g=v+a$ with $v\in V$ and $a\in A$. Tilings are a special case of normed factorizations, a type of factorization by subsets that was introduced by Hajos [Casopsis Puest Path. Rys., 74, (1949), pp. 157-162]. A tiling is said to be full rank if both $V$ and $A$ generate $G$. Cohen, Litsyn, Vardy, and Zemor [SIAM J. Discrete Math., 9 (1996), pp. 393-412] proved that any tiling of $\Z_2^n$ can be decomposed into full rank and trivial tilings. We generalize this decomposition from $\Z_2^n$ to all finite abelian groups. We also show how to generate larger full rank tilings from smaller ones, and give two sufficient conditions for a group to admit a full rank tiling, showing that many groups do admit them. In particular, we prove that if $p\geq 5$ is a prime and $n\geq 4$, then $\Z_p^n$ admits a full rank tiling. This bound on $n$ is tight for $5\leq p\leq 11$, and is conjectured to be tight for all primes $p$.

Factoring Finite Abelian Groups by Subsets with Maximal Span

We say that a finite abelian group has the Re´dei property if it does not admit factorization into two normalized subsets that both span the whole group. It will be shown that subgroups inherit the Re´dei property from the group. Then four constructions are described to exhibit groups without the Re´dei property. Using these we further narrow the list of $p$-groups that might have the Re´dei property.

Finite Abelian actions on surfaces

Edmonds showed that two free orientation preserving smooth actions φ 1 and φ 2 of a finite Abelian group G on a closed connected oriented smooth surface M are equivalent by an equivariant orientation preserving diffeomorphism iff they have the same bordism class [ M , φ 1 ] = [ M , φ 2 ] in the oriented bordism group Ω 2 ( G ) of the group G. In this paper, we compute the bordism class [ M , φ ] for any such action of G on M and we determine for a given M, the bordism classes in Ω 2 ( G ) that are representable by such actions of G on M. This will enable us to obtain a formula for the number of inequivalent such actions of G on M. We also determine the “weak” equivalence classes of such actions of G on M when all the p-Sylow subgroups of G are homocyclic (i.e. of the form ( Z / p α Z ) n ) .

On finite loops whose inner mapping groups are Abelian II

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.

Quasi-decompositions for Self-Small Abelian Groups#

We will prove a Krull-Schmidt type theorem for the quasi decompositions of a self-small abelian group of finite torsion free rank, which extends the classical result proved for finite rank torsion free abelian groups.

The Reconstructibility of Finite Abelian Groups

Given a subset $S$ of an abelian group $G$ and an integer $k\geq 1$, the $k$-deck of $S$ is the function that assigns to every $T\subseteq G$ with at most $k$ elements the number of elements $g\in G$ with $g+T\subseteq S$. The reconstruction problem for an abelian group $G$ asks for the minimal value of $k$ such that every subset $S$ of $G$ is determined, up to translation, by its $k$-deck. This minimal value is the set-reconstruction number $r_{\rm set}(G)$ of $G$; the corresponding value for multisets is the reconstruction number $r(G)$. Previous work had given bounds for the set-reconstruction number of cyclic groups: Alon, Caro, Krasikov and Roditty [1] showed that $r_{\rm set}({\mathbb{Z}}_n)<\log_2n$ and Radcliffe and Scott [15] that $r_{\rm set}({\mathbb{Z}}_n)<9\frac{\ln n}{\ln\ln n}$. We give a precise evaluation of $r(G)$ for all abelian groups $G$ and deduce that $r_{\rm set}({\mathbb{Z}}_n)\leq 6$.

On the Heyde Theorem for Finite Abelian Groups

It is well-known Heyde's characterization theorem for the Gaussian distribution on the real line: if ξ j are independent random variables, α j , β j are nonzero constants such that β i α ±β j α−1 j ≠ 0 for all i≠ j and the conditional distribution of L 2=β1 ξ1 + ··· + β n ξ n given L 1=α1 ξ1 + ··· +α n ξ n is symmetric, then all random variables ξ j are Gaussian. We prove some analogs of this theorem, assuming that independent random variables take on values in a finite Abelian group X and the coefficients α j ,β j are automorphisms of X.

Covering a Finite Abelian Group by Subset Sums

Let G be an abelian group of order n. The critical number c(G) of G is the smallest s such that the subset sums set Σ(S) covers all G for eachs ubset S⊂G\{0} of cardinality |S|≥s. It has been recently proved that, if p is the smallest prime dividing n and n/p is composite, then c(G)=|G|/p+p−2, thus establishing a conjecture of Diderrich.We characterize the critical sets with |S|=|G|/p+p−3 and Σ(S)=G, where p≥3 is the smallest prime dividing n, n/p is composite and n≥7p2+3p.We also extend a result of Diderrichan d Mann by proving that, for n≥67, |S|≥n/3+2 and S=G imply Σ(S)=G. Sets of cardinality $${\left| S \right|} \geqslant \frac{{n + 11}} {4}$$ for which Σ(S) =G are also characterized when n≥183, the smallest prime p dividing n is odd and n/p is composite. Finally we obtain a necessary and sufficient condition for the equality Σ(G)=G to hold when |S|≥n/(p+2)+p, where p≥5, n/p is composite and n≥15p2.

On the enumeration of finite Abelian and solvable groups

Let a( n) be the number of nonisomorphic abelian groups of order n. We obtain a short interval result for the local density of a( n). More generally, we get short interval version of results of Ivić on the local density of prime independent multiplicative functions. Also we prove a short interval version of the theorem of Erdös and Szekeres on the summatory function of a( n) and the theorem of Greenberg and Newman on the enumeration of a certain type of finite solvable groups.

On the Fundamental Theorem of Finite Abelian Groups

On the Fundamental Theorem of Finite Abelian Groups

On the Fundamental Theorem of Finite Abelian Groups

(2003). On the Fundamental Theorem of Finite Abelian Groups. The American Mathematical Monthly: Vol. 110, No. 2, pp. 153-154.

Subquadratic Growth of the Averaged Dehn Function for Free Abelian Groups

For finite rank free abelian groups with the standard presentation, the averaged Dehn function is proved to be subquadratic.

On finite loops whose inner mapping groups are Abelian

Loops are nonassociative algebras which can be investigated by using their multiplication groups and inner mapping groups If the inner mapping group of a loop is finite and Abelian, then the multiplication group is a solvable group. It is clear that not all finite Abelian groups can occur as inner mapping groups of loops. In this paper we show that certain finite Abelian groups with a special structure are not isomorphic to inner mapping groups of finite loops. We use our results and show how to construct solvable groups which are not isomorphic to multiplication groups of loops.

On Zeta-Functions of Finite Abelian Groups

In the paper, the zeta-functions of finite Abelian groups of rank 2 and 3 are considered. One- and two-dimensional limit theorems for these zeta-functions on the complex plane and in the space of meromorphic functions are obtained.