Minimal rank of abelian group matrices

The minimal rank of abelian group matrices with positive integral entries is determined.The corresponding problem for circulant matrices have been investigated by Ingleton and more recently by Shiu-Ma-Fang. Our work can be viewed as a generalization of their results, since a group matrix becomes circulant when the group is cyclic.     

Basic subgroups in modular abelian group algebras

Suppose F is a perfect field of char F = p ≠ 0 and G is an arbitrary abelian multiplicative group with a p-basic subgroup B and p-component G _p . Let FG be the group algebra with normed group of all units V(FG) and its Sylow p-subgroup S(FG), and let I _p (FG; B) be the nilradical of the relative augmentation ideal I(FG; B) of FG with respect to B. The main results that motivate this article are that 1 + I _p (FG; B) is basic in S(FG), and B(1 + I _p (FG; B)) is p-basic in V(FG) provided G is p-mixed. These achievements extend in some way a result of N. Nachev (1996) in Houston J. Math. when G is p-primary. Thus the problem of obtaining a (p-)basic subgroup in FG is completely resolved provided that the field F is perfect. Moreover, it is shown that G _p (1 + I _p (FG; B))/G _p is basic in S(FG)/G _p , and G(1 + I _p (FG; B))/G is basic in V(FG)/G provided G is p-mixed. As consequences, S(FG) and S(FG)/G _p are both starred or divisible groups. All of the listed assertions enlarge in a new aspect affirmations established by us in Czechoslovak Math. J. (2002), Math. Bohemica (2004) and Math. Slovaca (2005) as well.     

Minimal faithful representations of abelian Jordan algebras and Lie superalgebras

Over an algebraically closed field of characteristic zero, all the abelian subalgebras of the maximal dimension are classified for any special Jordan algebra. As a consequence, the minimal dimension of the faithful representations of any finite-dimensional abelian Jordan algebra is determined and the minimal faithful representations are classified for the so-called nice abelian Jordan algebras. The same is done for the purely odd Lie superalgebras.     

Derivations on group algebras of a locally compact abelian group

Monatshefte für Mathematik - Let G be a locally compact abelian group. In this paper, we study derivations on the Banach algebra $$L_0^infty (G)^*$$ . We prove that any derivation on...     

Minimal pseudocompact group topologies on free abelian groups

A Hausdorff topological group G is minimal if every continuous isomorphism f:G→H between G and a Hausdorff topological group H is open. Significantly strengthening a 1981 result of Stoyanov, we prove the following theorem: For every infinite minimal abelian group G there exists a sequence of cardinals such that

     

Quasitriangular Hopf algebras whose group-like elements form an abelian group

In this paper we prove some properties of the set of group-like elements of , , for a pointed minimal quasitriangular Hopf algebra over a field of characteristic 0, and for a pointed quasitriangular Hopf algebra which is indecomposable as a coalgebra. We first show that over a field of characteristic 0, for any pointed minimal quasitriangular Hopf algebra , is abelian. We show further that if is a quasitriangular Hopf algebra which is indecomposable as a coalgebra, then is contained in , the minimal quasitriangular Hopf algebra contained in . As a result, one gets that over a field of characteristic 0, a pointed indecomposable quasitriangular Hopf algebra has a finite abelian group of group-like elements.

     

On the isomorphism of real group algebras of abelian groups

Necessary and sufficient conditions for the isomorphism of real group algebras of arbitrary Abelian groups are found. Translated from Matematicheskie Zametki, Vol. 21, No. 2, pp. 229–238, February, 1977.     

Commutative group algebras of -summable abelian groups

In this note we study the commutative modular and semisimple group rings of -summable abelian -groups, which group class was introduced by R. Linton and Ch. Megibben. It is proved that is -summable if and only if is -summable, provided is an abelian group and is a commutative ring with 1 of prime characteristic , having a trivial nilradical. If is a -summable -group and the group algebras and over a field of characteristic are -isomorphic, then is a -summable -group, too. In particular provided is totally projective of a countable length.

Moreover, when is a first kind field with respect to and is -torsion, is -summable if and only if is a direct sum of cyclic groups.

     

On perfectt-shift codes in abelian groups

LetG be a finite abelian group,t a positive integer. Thet-shift sphere with centerx G is the setS _t (x)={±ix|i=1,...,t}. At-shift code is a subsetX ofG such that the setsS _t (x) (x X) have size 2t and are disjoint. Clearly, the sphere packing bound: 2t|X|+1|G| holds for anyt-shift codeX. Aperfect t-shift code is at-shift codeX with 2t|X|+1=|G|. A necessary and sufficient condition for the existence of a perfectt-shift code in a finite abelian group is known fort-1, 2. In this paper, we determine finite abelian groups in which there exists a perfectt-shift code fort=3, 4.This research was completed during the author's visit at the Institute for System Analysis, Moscow, as a Heizaemon Honda fellow of the Japan Association for Mathematical Sciences.     

On weights in duadic abelian codes

In this note, we prove that if C is a duadic binary abelian code with splitting μ=μ₋₁ and the minimum odd weight of C satisfies d²−d+1≠n, then d(d−1)n+11. We show by an example that this bound is sharp. A series of open problems on this subject are proposed.     

On abelian difference set codes

In this paper we determine the ranks of the incidence matrices that belong to the following types of difference sets: Twin prime power difference sets, biquadratic residues and biquadratic residues with 0. We also prove a conjecture of Assmus and Key on the code generated by the hyperovals of PG(2, q).     

Minimal relation algebras

This paper is concerned with the covers of the atoms in the lattice of varieties of relation algebras. Aminimal relation algebra is one that is simple and generates such a subvariety. The main result we prove is that there are exactly three finite minimal relation algebras that aretotally symmetric (i.e., satisfy the identitiesx=x andx x; x). We also give an example of an infinite minimal totally symmetric relation algebra, and some results about other subvarieties.Dedicated to Bjarni Jonsson on his 70th birthdayPresented by G. McNulty.