Factoring by subsets of cardinality of prime power

Rédei's theorem asserts that if a finite abelian group is expressed as a direct product of subsets of prime cardinality, then at least one of the factors must be periodic. (A periodic subset is a direct product of some subset and a nontrivial subgroup.) A. D. Sands proved that if a finite cyclic group is the direct product of subsets each of which has cardinality that is a power of a prime, then at least one of the factors is periodic. We prove that the same conclusion holds if a general finite abelian group is factored as a direct product of cyclic subsets of prime cardinalities and general subsets of cardinalities that are powers of primes provided that the components of the group corresponding to these latter primes are cyclic.     

Explicit determination of certain minimal constabelian codes

Constabelian codes can be viewed as ideals in twisted group algebras over finite fields. In this paper we study decomposition of semisimple twisted group algebras of finite abelian groups and prove results regarding complete determination of a full set of primitive orthogonal idempotents in such algebras. We also explicitly determine complete sets of primitive orthogonal idempotents of twisted group algebras of finite cyclic and abelian p-groups. We also describe methods of determining complete set of primitive idempotents of abelian groups whose orders are divisible by more than one prime and give concrete (numerical) examples of minimal constabelian codes, illustrating the above mentioned results.     

A new upper bound for the cross number of finite abelian groups

In this paper, building among others on earlier works by U. Krause and C. Zahlten (dealing with the case of cyclic groups), we obtain a new upper bound for the little cross number valid in the general case of arbitrary finite abelian groups. Given a finite abelian group, this upper bound appears to depend only on the rank and the number of distinct prime divisors of the exponent. The main theorem of this paper allows us, among other consequences, to prove that a classical conjecture concerning the cross and little cross numbers of finite abelian groups holds asymptotically in at least two different directions.     

The duals of almost completely decomposable groups

In this paper, we classify the direct products of one-dimensional compact connected abelian groups by cardinal invariants dualizing Baer’s classification theorem of completely decomposable groups. Almost completely decomposable groups are finite rank torsion-free abelian groups which contain a completely decomposable group of finite index. An isomorphism theorem for their Pontrjagin dual groups is given by using the dual concept of a regulating subgroup.     

Functors between shape categories

Permutation groups of prime power degree are investigated here through the study of the corresponding group algebra of the set of all functions from the underlying set on which the permutation group acts to a finite field of characteristic p. For the case when the permutation group is of degree p² acting on a set consisting of the direct product of two elementary abelian p-groups, the structure of a minimal permutation module is obtained under certain conditions. The proofs do not depend on the recent classification results of finite simple groups.     

On the lower central series of the free nilpotent groups of finite rank

In their article, “On the derived subgroup of the free nilpotent groups of finite rank” R. D. Blyth, P. Moravec, and R. F. Morse describe the structure of the derived subgroup of a free nilpotent group of finite rank n as a direct product of a nonabelian group and a free abelian group, each with a minimal generating set of cardinality that is a given function of n. They apply this result to computing the nonabelian tensor squares of the free nilpotent groups of finite rank. We generalize their main result to investigate the structure of the other terms of the lower central series of a free nilpotent group of finite rank, each again described as a direct product of a nonabelian group and a free abelian group. In order to compute the ranks of the free abelian components and the size of minimal generating sets for the nonabelian components we introduce what we call weight partitions.     

On Purity and Quasiequality of Abelian Groups

We study the Cohn purity in an abelian group regarded as a left module over its endomorphism ring. We prove that if a finite rank torsion-free abelian group G is quasiequal to a direct sum in which all summands are purely simple modules over their endomorphism rings then the module _E(G) G is purely semisimple. This theorem makes it possible to construct abelian groups of any finite rank which are purely semisimple over their endomorphism rings and it reduces the problem of endopure semisimplicity of abelian groups to the same problem in the class of strongly indecomposable abelian groups.     

A Model-Theoretic Characterization of Countable Direct Sums of Finite Cyclic Groups #

We prove that an abelian group G is a countable direct sum of finite cyclic groups if and only if there exists a consistent existential theory Γ of abelian groups such that G is embeddable in every model of Γ.     

Hamiltonian circuits in Cayley graphs

The following result is proved: If either G is a finite abelian group or a semidirect product of a cyclic group of prime order by a finite abelian group of odd order, then every connected Cayley graph of G is hamiltonian.     

Direct summands of class groups

Let G be a finite abelian group, and F a global field of characteristic prime to the order of G. Then there exists a finite extension of F whose class group has a direct summand isomorphic to G.     

Transfer and Tate’s theorem

Let H be a subgroup of a finite group G, and assume that p is a prime that does not divide |G : H|. In favorable circumstances, one can use transfer theory to deduce that the largest abelian p-groups that occur as factor groups of G and of H are isomorphic. When this happens, Tate’s theorem guarantees that the largest not-necessarily-abelian p-groups that occur as factor groups of G and H are isomorphic. Known proofs of Tate’s theorem involve cohomology or character theory, but in this paper, a new elementary proof is given. It is also shown that the largest abelian p-factor group of G is always isomorphic to a direct factor of the largest abelian p-factor group of H. Received: 17 June 2008     

Scenery reconstruction on finite abelian groups

We consider the question of when a random walk on a finite abelian group with a given step distribution can be used to reconstruct a binary labeling of the elements of the group, up to a shift. Matzinger and Lember (2006) give a sufficient condition for reconstructability on cycles. While, as we show, this condition is not in general necessary, our main result is that it is necessary when the length of the cycle is prime and larger than 5, and the step distribution has only rational probabilities. We extend this result to other abelian groups.     

Constructions Related to the Redei Property of Groups

We say that a finite abelian group does not have the Rédeiproperty if it can be expressed as a direct product of two ofits subsets such that both subsets contain the identity elementand both subsets span the whole group. It will be shown thatonly a small fraction of the finite abelian groups can havethe Rédei property. For groups of odd order an explicitlist of the possible exceptions is compiled.     

On one-sided stabilizers of subsets of finite groups

It was initiated by the second author to investigate in which groups the left and right stabilizers of subsets have equal order. First we prove that if the left and right stabilizers of sets of prime power size are equal order then the group is supersolvable. We also characterize those 2-groups which satisfy this property for p = 2. We show that if in a finite group, the left and right stabilizers of sets of prime power size have equal order, then the commutator subgroup is abelian. Finally we characterize hamiltonian groups with the help of one-sided stabilizers. Received: 18 April 2005; revised 11 May 2005     

Finite groups with planar subgroup lattices

It is natural to ask when a group has a planar Hasse lattice or more generally when its subgroup graph is planar. In this paper, we completely answer this question for finite groups. We analyze abelian groups, p-groups, solvable groups, and nonsolvable groups in turn. We find seven infinite families (four depending on two parameters, one on three, two on four), and three “sporadic” groups. In particular, we show that no nonabelian group whose order has three distinct prime factors can be planar.     

A new result on difference sets with -1 as multiplier

In this paper we study finite abelian groups admitting a difference set with multiplier -1. In these groups we have that each integer, which is relatively prime to the group order, is a multiplier (see [1] and Section 1 of this paper).About the arithmetical structure, there is an interesting result of Jungnickel [3] on primes dividing the order n of the corresponding design. Here we prove (see Theorem 2.1) that each odd prime divisor of the order v of the group divides n. The proof of Theorem 2.1 rests on character computations and is motivated by [5].     

On r*-sequenceability and symmetric harmoniousness of groups. II

In this paper we investigate symmetric harmoniousness of groups and connections of this concept to the R*-sequenceability of groups. We prove that, under suitable assumptions, the direct product of a symmetric harmonious group with a group that is R*-sequenceable is R*-sequenceable; we discuss the symmetric harmoniousness of abelian and of nilpotent groups; we also prove that, for a fixed odd prime p, all but possibly finitely many of the nonabelian groups of order pq (q prime, q ≡ 1 (mod p)) are symmetric harmonious. © 1995 John Wiley & Sons, Inc.     

Malcev presentations for subsemigroups of direct products of coherent groups

The direct product of a free group and a polycyclic group is known to be coherent. This paper shows that every finitely generated subsemigroup of the direct product of a virtually free group and an abelian group admits a finite Malcev presentation. (A Malcev presentation is a presentation of a special type for a semigroup that embeds into a group. A group is virtually free if it contains a free subgroup of finite index.) By considering the direct product of two free semigroups, it is also shown that polycyclic groups, unlike nilpotent groups, can contain finitely generated subsemigroups that do not admit finite Malcev presentations.     

Finite metacyclic groups as active sums of cyclic subgroups

The notion of active sum provides an analogue for groups of what the direct sum is for abelian groups. One natural question then is which groups are the active sum of a family of cyclic subgroups. Many groups have been found to give a positive answer to this question, while the case of finite metacyclic groups remained unknown. In this note we show that every finite metacyclic group can be recovered as the active sum of a discrete family of cyclic subgroups.     

Some basic properties of planar Singer groups

A planar Singer group is a collineation group of a finite (in this article) projective plane acting regularly on the points of the plane. Theorem 1 gives a characterization of abelian planar Singer groups. This leads to a necessary and sufficient condition for an inner automorphism to be a multiplier. The Sylow 2-structure of a multiplier group and some of its consequences are given in Theorem 3. One important result in studying multipliers of an abelian Singer group is the existence of a common fixed line. We extend this to an arbitrary planar Singer group in Theorem 4. Theorem 5 studies the order of an abelian group of multiplers. If this order equals to the order of the plane plus 1, then the number of points of the plane is a prime. If this order is odd, then it is at most the planar order plus 1.Partially supported by a NSA grant.