Abelian Group Structure Research Articles

The structure of finite abelian n-ary groups

AbstractThe structure of finite abelian n-ary groups is completely described: it is reduced to the direct product of primary abelian semicyclic n-ary groups up to isomorphism.

Multilevel Channel Polarization for Arbitrary Discrete Memoryless Channels

It is shown that polar codes, with their original (u,u+v) kernel, achieve the symmetric capacity of discrete memoryless channels with arbitrary input alphabet sizes. It is shown that in general, channel polarization happens in several, rather than only two levels so that the synthesized channels are either useless, perfect or “partially perfect.” Any subset of the channel input alphabet which is closed under addition induces a coset partition of the alphabet through its shifts. For any such partition of the input alphabet, there exists a corresponding partially perfect channel whose outputs uniquely determine the coset to which the channel input belongs. By a slight modification of the encoding and decoding rules, it is shown that perfect transmission of certain information symbols over partially perfect channels is possible. Our result is general regarding both the cardinality and the algebraic structure of the channel input alphabet; i.e., we show that for any channel input alphabet size and any Abelian group structure on the alphabet, polar codes are optimal. Due to the modifications, we make to the encoding rule of polar codes, the constructed codes fall into a larger class of structured codes called nested group codes.

Algebraic and topological structures on the set of mean functions and generalization of the AGM mean

In this paper, we present new structures and results on the set MD of mean functions on a given symmetric domain D of R 2 . First, we construct on MD a structure of abelian group in which the neutral element is simply the Arithmetic mean; then we study some symmetries in that group. Next, we construct on MD a structure of metric space under which MD is nothing else the closed ball with center the Arithmetic mean and radius 1=2. We show in particular that the Geometric and Harmonic means lie in the border of MD . Finally, we give two important theorems generalizing the construction of the AGM mean. Roughly speaking, those theorems show that for any two given means M1 and M2, which satisfy some regular conditions, there exists a unique mean M satisfying the functional equation: M(M1;M2) = M . MSC: 20K99; 54E35; 39B22; 39B12.

The equations satisfied by GGS-groups and the abelian group structure of the Gupta–Sidki group

The group Γ of p-adic automorphisms of the p-adic rooted tree can naively be identified with the p-group FpN, using pointwise addition + in the portraits of the automorphisms. We prove that the operation + is also internal for the Gupta–Sidki group, as well as for some other discrete subgroups of Γ. In order to get this, we introduce the notion of equation, or pattern, for subgroups of Γ, and we describe all equations for these groups.

Computing Gröbner bases of pure binomial ideals via submodules of [formula omitted

A binomial ideal is an ideal of the polynomial ring which is generated by binomials. In a previous paper, we gave a correspondence between pure saturated binomial ideals of K[x1,…,xn] and submodules of Zn and we showed that it is possible to construct a theory of Gröbner bases for submodules of Zn. As a consequence, it is possible to follow alternative strategies for the computation of Gröbner bases of submodules of Zn (and hence of binomial ideals) which avoid the use of Buchberger algorithm. In the present paper, we show that a Gröbner basis of a Z-module M⊆Zn of rank m lies into a finite set of cones of Zm which cover a half-space of Zm. More precisely, in each of these cones C, we can find a suitable subset Y(C) which has the structure of a finite abelian group and such that a Gröbner basis of the module M (and hence of the pure saturated binomial ideal represented by M) is described using the elements of the groups Y(C) together with the generators of the cones.

The additivity of theρ–invariant and periodicity in topological surgery

For a closed topological manifold M with dim (M) >= 5 the topological structure set S(M) admits an abelian group structure which may be identified with the algebraic structure group of M as defined by Ranicki. If dim (M) = 2d-1, M is oriented and M is equipped with a map to the classifying space of a finite group G, then the reduced rho-invariant defines a function, \wrho : S(M) \to \QQ R_{hat G}^{(-1)^d}, to a certain sub-quotient of the complex representation ring of G. We show that the function \wrho is a homomorphism when 2d-1 >= 5. Along the way we give a detailed proof that a geometrically defined map due to Cappell and Weinberger realises the 8-fold Siebenmann periodicity map in topological surgery.

On finite loops with nilpotent inner mapping groups

We shall show how the nilpotency class of a finite loop Q is determined by the properties of a nilpotent inner mapping group. We also show that a classical result by Baer on the structure of abelian finite capable groups holds for Moufang loops of odd order.

The structure of abelian groups supporting a number system (extended abstract)

The structure of abelian groups supporting a number system (extended abstract)

Numerical evidence on the uniform distribution of power residues for elliptic curves

Elliptic curves are fascinating mathematical objects which occupy the intersection of number theory, algebra, and geometry. An elliptic curve is an algebraic variety upon which an abelian group structure can be imposed. By considering the ring of endomorphisms of an elliptic curve, a property called complex multiplication may be defined, which some elliptic curves possess while others do not. Given an elliptic curve E and a prime p, denote by N p the number of points on E over the finite field F p. It has been conjectured that given an elliptic curve E without complex multiplication and any modulus M, the primes for which N p is a square modulo p are uniformly distributed among the residue classes modulo M. This paper offers numerical evidence in support of this conjecture.

Abelian torsion groups with a countably compact group topology

Comfort and Remus [W.W. Comfort, D. Remus, Abelian torsion groups with a pseudocompact group topology, Forum Math. 6 (3) (1994) 323–337] characterized algebraically the Abelian torsion groups that admit a pseudocompact group topology using the Ulm–Kaplansky invariants. We show, under a condition weaker than the Generalized Continuum Hypothesis, that an Abelian torsion group (of any cardinality) admits a pseudocompact group topology if and only if it admits a countably compact group topology. Dikranjan and Tkachenko [D. Dikranjan, M. Tkachenko, Algebraic structure of small countably compact Abelian groups, Forum Math. 15 (6) (2003) 811–837], and Dikranjan and Shakhmatov [D. Dikranjan, D. Shakhmatov, Forcing hereditarily separable compact-like group topologies on Abelian groups, Topology Appl. 151 (1–3) (2005) 2–54] showed this equivalence for groups of cardinality not greater than 2 c . We also show, from the existence of a selective ultrafilter, that there are countably compact groups without non-trivial convergent sequences of cardinality κ ω , for any infinite cardinal κ. In particular, it is consistent that for every cardinal κ there are countably compact groups without non-trivial convergent sequences whose weight λ has countable cofinality and λ > κ .

On the Intersection of ${{\BBZ}_{2}{\BBZ}_{4}}$-Additive Hadamard Codes

The intersection structure for Z 2 Z 4-additive Hadamard codes is investigated. For any two of these codes C 1 and C 2, the Abelian group structure of the intersection C 1 capC 2 is characterized. The parameters of this Abelian group structure corresponding to the intersection codes are computed, establishing lower and upper bounds for them. Constructions are given of codes whose intersection has any parameters between these bounds.

A Composition Formula for Manifold Structures

The structure set STOP (M) of an n-dimensional topological manifold M for n > 5 has a homotopy invariant functorial abelian group structure, by the algebraic version of the Browder-Novikov-Sullivan-Wall surgery theory. An element (N, f) ∈ STOP (M) is an equivalence class of n-dimensional manifolds N with a homotopy equivalence f : N → M . The composition formula is that (P, fg) = (N, f) + f∗(P, g) ∈ STOP (M) for homotopy equivalences g : P → N , f : N → M . The formula is required for a paper of Kreck and Luck.

On the Intersection of $ {\BBZ }_{2} {\BBZ }_{4}$-Additive Perfect Codes

The intersection problem for Z2Z4-additive (extended and nonextended) perfect codes, i.e., which are the possibilities for the number of codewords in the intersection of two Z2Z4-additive codes C1 and C2 of the same length, is investigated. Lower and upper bounds for the intersection number are computed and, for any value between these bounds, codes which have this given intersection value are constructed. For all these Z2Z4-additive codes C1 and C2, the abelian group structure of the intersection codes C1 cap C2 is characterized. The parameters of this Abelian group structure corresponding to the intersection codes are computed and lower and upper bounds for these parameters are established. Finally, for all possible parameters between these bounds, constructions of codes with these parameters for their intersections are given.

A group structure on squares

We show that there is an abelian group structure on the orbit set of squares of unimodular rows of length n over a commutative ring of stable dimension d when d = 2 n - 3, n odd and also an abelian group structure on the orbit set of fourth powers of unimodular rows of length n over a commutative ring of stable dimension d when d = 2n - 3, n even.

Complexité bilinéaire de la multiplication dans des petits corps finis

In this Note, we improve a result of Shokrollahi who has applied the algorithm of D.V. Chudnovsky and G.V. Chudnovsky to algebraic curves of genus one. More precisely, from the Abelian group structure of the set of rational points on elliptic curves, we show that, if 1 2 q + 1 < n ⩽ 1 2 ( q + 1 + ϵ ( q ) ) , then the bilinear complexity of the multiplication in all extensions F q n is equal to 2 n. To cite this article: J. Chaumine, C. R. Acad. Sci. Paris, Ser. I 343 (2006).

Multiple zeta functions and asymptotic structure of free Abelian groups of finite rank

We define the multiple zeta function of the free Abelian group Z d as ζ Z d ( s 1 , … , s d ) = ∑ | Z d : H | < ∞ α 1 ( H ) − s 1 ⋯ α d ( H ) − s d , where Z d / H ≅ C α 1 ( H ) ⊕ ⋯ ⊕ C α d ( H ) is the canonical decomposition into cyclic factors, and α i + 1 ( H ) | α i ( H ) for i = 1 , … , d − 1 . As the main result, we compute this function, find the region of absolute convergence, and study its analytic continuation. Our result allows us to describe an asymptotic structure of a “random” finite factor group Z d / H as follows. For a subgroup of finite index H ⊆ Z d , consider the order of the product of the canonical cyclic factors except the largest one, σ ( H ) = α 2 ( H ) ⋯ α d ( H ) . Fix n ∈ N , and let σ n ( d ) be the arithmetic mean of σ ( H ) over all subgroups H ⊆ Z d of index at most n . We prove that there exists a limit lim n → ∞ σ n ( d ) , and this number is bounded by 1.243, for all ranks d ≥ 1 . In this sense, a random finite factor group Z d / H is very close to a cyclic group. We also compute the zeta function that enumerates cyclic finite factor groups. This result allows us to amend our observation that a random finite factor group Z d / H is close to a cyclic group in the following way. Consider all subgroups H ⊆ Z d of index at most n , and let τ n ( d ) be the share of the subgroups such that Z d / H is cyclic. We compute τ ( d ) = lim n → ∞ τ n ( d ) , which can be considered as the probability that Z d / H is cyclic, and we show that τ ( d ) ≥ 0.8469 … for all d ≥ 1 . Also, we apply our results to study similar questions for free modules of finite rank over finitely generated Dedekind domains.

Strong Limit Power Functions

A new C*-algebra of strong limit power functions is proposed. The Gelfand space of the C*-algebra is endowed with an Abelian compact group structure. As applications of this, Fourier analysis and the Bochner-Fejer approximation are carried out for a strong limit power function. Finally, the functions are extended to more general cases and their properties are investigated in that settings.

A FEW HOMOLOGICAL CHARACTERIZATIONS OF COMPACT SEMISIMPLE RINGS

Fix any compact ring R with identity. We associate to R the following categories of topological R-modules: (i) R𝔇 (𝔇R) the category of all discrete topological left (right) R-modules; (ii) Rℭ (ℭR) the category of all compact left (right) R-modules. We have introduced the following notions (analogous with classical notions of module theory): (i) the tensor product [Formula: see text] of A ∈ ℭR and B ∈Rℭ ([Formula: see text] has a structure of a compact Abelian group); (ii) a topologically semisimple module; (iii) a compact topologically flat module. We give a characterization of compact semisimple rings by using of flat modules.

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.

Computing the structure of a finite abelian group

We present an algorithm that computes the structure of a finite abelian group G from a generating system M. The algorithm executes O(|M|√|G|) group operations and stores O(√|G|) group elements.