Abelian Group Of Order Research Articles

Hamiltonian Decompositions of Cayley Graphs on Abelian Groups of Odd Order

Alspach has conjectured that any 2k-regular connected Cayley graphcay(A, S) on a finite abelian groupAcan be decomposed intokhamiltonian cycles. In this paper, the conjecture is shown to be true ifS={s1,s2, …,sk} is a minimal generating set of an abelian groupAof odd order (where a generating setSof a groupGisminimalif no proper subset ofGcan generateG).

On Weighted Sequence Sums

The main result of this paper has the following consequence. Let G be an abelian group of order n. Let {xi: 1 ≤ 2n − 1} be a family of elements of G and let {wi: 1 ≤ i ≤ n − 1} be a family of integers prime relative to n. Then there is a permutation & of [1,2n − 1] such thatApplying this result with wi = 1 for all i, one obtains the Erdős–Ginzburg–Ziv Theorem.

Addition Theorems for Finite Abelian Groups

Let a1, ..., ak be a sequence of elements in an Abelian group of order n (repetition allowed). In this paper, we give two sufficient conditions such that an element g ∈ G can be written in the form g = ai1 + ai2 + · · · + ain, 1 ≤ i1 < i2 < · · · <in ≤ k, if and only if for every b ∈ G, g can be written in the form g = (b + aj1) + · · · + (b + ajl(b), 1 ≤ j1 < · · · jl(b) ≤ k, 1 ≤ l(b) ≤ k. As an application of this result, we improve a result of Olson.

On the number of finite non-isomorphic Abelian groups in short intervals

Let a(n) denote the number of non-isomorphic Abelian groups of order n. It is well-known thatfor a natural number k we defineand

On the Sherk plane

In this paper, we characterize the Sherk plane as the only non-semifield translation plane of order 27 that admits an abelian group of order 27 in the translation complement.

On ??-groups and fields

Let κ:=ℵα. The following are known: two ηα-sets of power κ are isomorphic. Let α>0. Two ordered divisible Abelian groups that are ηα-sets of power κ are isomorphic, two real closed fields that are ηα-sets of power κ are isomorphic. The following is shown: (1) there exist 2κ nonisomorphic ordered Abelian groups (respectively ordered fields) that are ηα-sets of power κ; (2) there exist 2κ nonisomorphic ordered divisible Abelian groups (respectively real closed fields) of power κ all having the same order type; (3) there exist 2 nonisomorphic ordered divisible Abelian groups (respectively real closed fields) that are ηα-sets having the same order type.

New constructions of divisible designs

A construction is given for a ( p 2 a ( p+1), p 2, p 2 a+1 ( p+1), p 2 a+1 , p 2 a ( p+1)) ( p a prime) divisible difference set in the group H× Z 2 pa+1 where H is any abelian group of order p+1. This can be used to generate a symmetric semi-regular divisible design; this is a new set of parameters for λ 1≠0, and those are fairly rare. We also give a construction for a ( p a−1 + p a−2 +…+ p+2, p a+2 , p a ( p a + p a−1 +…+ p+1), p a ( p a−1 +…+ p+1), p a−1 ( p a +…+ p 2+2)) divisible difference set in the group H× Z p2 × Z a p . This is another new set of parameters, and it corresponds to a symmetric regular divisible design. For p=2, these parameters have λ 1= λ 2, and this corresponds to the parameters for the ordinary Menon difference sets.

A generalization of Kraemer's result on difference sets

Kraemer has shown that every abelian group of order 2 2 d + 2 with exponent less than 2 2 d + 3 has a difference set. Generalizing this result, we show that any non-abelian group with a central subgroup of size 2 d + 1 together with an exponent-like condition will have a difference set.

Transversals in the Cayley Tables of the Non-cyclic Groups of Order 8

In the course of a recent investigation by the author [1] it was noted that the number of complete mappings (transversals) of each of the non-cyclic groups of order 8 was 364. Furthermore, it was found that both of the non-abelian groups of order 8 possessed exactly 128 near-complete mappings. These surprising results motivated an investigation into why this was the case. In the present paper we show why both of the non-abelian groups of order 8 possess the same number of complete and near-complete mappings. It is also shown why both of the non-cyclic abelian groups of order 8 possess the same number of complete mappings.

Topological invariants of real algebraic actions

In this paper we prove the Fixed Point Conjecture for odd order abelian groups, and we construct algebraically Smith equivalent representations. Furthermore, we show that certain nonlinear phenomena in smooth transformation groups also occur in real algebraic transformation groups.

An Example of Two Nonisomorphic Countable ordered Abelian Groups with Isomorphic Lexicographical Squares

An Example of Two Nonisomorphic Countable ordered Abelian Groups with Isomorphic Lexicographical Squares

Cancellation and absorption of lexicographic powers of totally ordered Abelian groups

Let G and H be totally ordered Abelian groups such that, for some integer k, the lexicographic powers G k and H k are isomorphic (as ordered groups). It was proved by F. Oger that G and H need not be isomorphic. We show here that they are whenever G is either divisible or ω1 -saturated (and in a few more cases). Our proof relies on a general technique which we also use to prove that G and H must be elementary equivalent as ordered groups (a fact also proved by F. Delon and F. Lucas) and isomorphic as chains. The same technique applies to the question of whether G and H should be isomorphic as groups, but, in this last case, no hint about a possible negative answer seems available.

On 1-factorizability of Cayley graphs

In this paper we consider the existence of a 1-factorization of undirected Cayley graphs of groups of even order. We show that a 1-factorization exists for all generating sets for even order abelian groups, dihedral groups, and dicyclic groups and for all minimal generating sets for even order nilpotent groups and for D m × Z n . We also derive other results that are useful in considering specific Cayley graphs. These results support the conjecture that all Cayley graphs of groups of even order are 1-factorizable. If this is not the case the same result may hold for minimal generating sets.

Automorphisms of G-Azumaya Algebras

Let R be a commutative ring, G a finite abelian group of order n and exponent m, and assume n is a unit in R. In [10], F. W. Long defined a generalized Brauer group, BD(R, G), of algebras with a G-action and G-grading, whose elements are equivalence classes of G-Azumaya algebras. In this paper we investigate the automorphisms of a G-Azumaya algebra A and prove that if Picm(R) is trivial, then these automorphisms are all, in some sense, inner.In fact, each of these “inner” automorphisms can be written as the composition of an inner automorphism in the usual sense and a “linear“ automorphism, i.e., an automorphism of the typewith r(σ) a unit in R. We then use these results to show that the group of gradings of the centre of a G-Azumaya algebra A is a direct summand of G, and thus if G is cyclic of order pr, A is the (smash) product of a commutative and a central G-Azumaya algebra.

A note on finite semifield planes that admit affine homologies

Let π be a translation plane of order n that admits an abelian group of order n in its translation complement. If π admits an affine homology then it is a semifield plane.

Smith equivalence of representations for odd order cyclic groups

THEOREM A. There exist odd order cyclic groups which have Smith equivalent but nonisomorphic representations. After a short background discussion, we shall give a list of conditions (1.1) which are important in the development of this paper and state more precise versions of our results. Theorem B provides nonisomorphic Smith equivalent representations if one has nonisomorphic representations of G which satisfy (1.1). Corollary C shows the existence of infinitely many groups which have nonisomorphic representations which satisfy (1.1). The orders of some cyclic groups of this type are described by Lemma 2.5 and Corollary 2.6. Taken together, Theorem B and Corollary C prove Theorem A. The concept of Smith equivalence is based upon a question posed by P. A. Smith in a 1960 survey article [32, p. 4061. In our terminology the question is whether Smith equivalent representations are linearly isomorphic. Atiyah-Bott [3] and Milnor [17] established an affirmative answer for cyclic groups of odd prime power order and for representations with semifree action. By definition a group acts semifreely if each point is left fixed by only the trivial group element or by each group element. An extension of this work by Sanchez [28] implied the answer also to be affirmative if G is cyclic of order pq where p and q are odd primes. Bredon [6] showed for 2-groups that Smith equivalent representations are isomorphic if their dimension is large in comparison to the order of the group. The first negative answers to Smith’s question were established by the second author for odd order abelian groups with at least four noncyclic Sylow subgroups. See [23], [24]. Subsequently a number of papers have established a negative answer for even order groups, but the case of odd order cyclic groups has remained open. For cyclic groups of even order see the papers [7], [25], [ 111, [3 13, For noncyclic abelian groups and nonabelian 2-groups, see [34] and [S]. There are many interesting results and ideas in these papers. Results and techniques used in answering Smith’s question varj markedly according to the parity of the order of G. In Remark 1.2(ii) we explain the difference of the geometric and number theoretic treatment of even and odd order groups as it arises from the Atiyah-Bott extension of the Lefschetz Fixed Point theorem. One interesting question, untouched here, is to relate the differentiable structure of a homotopy sphere Z and the isotropy representations of an action of G on Z with exactly two fixed points. Some general remarks on this appear in the problem section of the forthcoming proceedings of the 1983 conference on group actions at Boulder [393. It can be shown that the homotopy spheres with group actions that are constructed by our method are all standard spheres. The methods of Schultz (e.g. in [29] and other papers) are relevant to realizing these actions on exotic spheres.

The Theory of Ordered Abelian Groups does not have the Independence Property

The Theory of Ordered Abelian Groups does not have the Independence Property

Orderings in locally compact Abelian groups and the theorem of F. and M. Riesz

Background (1·1). Ordered Abelian groups have been studied for nearly a century. Since the early 1950's, it has been recognized that orderings in locally compact Abelian groups can play an important rôle in harmonic analysis on such groups. In this paper we study orderings, especially in topological Abelian groups with either topological or measure-theoretic properties, obtaining nearly a complete classification of such orderings. We then apply these results to determine the limitations of the celebrated theorem of F. and M. Riesz on such groups.

Addition sets in a finite group

Addition sets with parameters ( υ, k, λ, α) are defined in a finite group G of order υ, where α: G → G is a homomorphism or anti-homomorphism. It is shown that addition sets in a finite group have similar properties to that of ( υ, k, λ, g) cyclic addition sets. The case α = I, where I: G → G is the identity automorphism, is studied and it is shown that no ( υ, k, λ, I) group addition sets exist in an Abelian group of order υ, where ether υ is odd, υ≡2 (mod 4), or in certain cases when υ≡0 (mod 4). Many examples of group addition sets in both Abelian and non-Abelian groups are provided.

Partitions of groups and complete mappings

Let G be an abelian group of order n and let k be a divisor of n — 1. We wish to determine whether there exists a complete mapping of G which fixes the identity element and permutes the remaining elements as a product of disjoint ^-cycles. We conjecture that if G has trivial or noncyclic Sylow 2-subgroup then such a mapping exists for every divisor k of n — 1. Several special cases of the conjecture are proved in this paper. We also prove that a necessary condition for the existence of such a map holds for every k when G is cyclic. 1* Introduction* A complete mapping of a group G is defined to be a bijection φ: G -> G such that the mapping θ: g —> g~λφ(g) is also bijective. (Some authors refer to θ, rather than φ, as the complete mapping.) If the permutation ( ι 2 ''' * J is a complete map