On infinite tournaments with regular automorphism groups

Atournament regular representation (TRR) of an abstract groupG is a tournamentT whose automorphism group is isomorphic toG and is a regular permutation group on the vertices ofT. L. Babai and W. Imrich have shown that every finite group of odd order exceptZ ₃ ×Z ₃ admits a TRR. In the present paper we give several sufficient conditions for an infinite groupG with no element of order 2 to admit a TRR. Among these are the following: (1)G is a cyclic extension byZ of a finitely generated group; (2)G is a cyclic extension byZ _2n+1 of any group admitting a TRR; (3)G is a finitely generated abelian group; (4)G is a countably generated abelian group whose torsion subgroup is finite.     

Finite digraphs with given regular automorphism groups

With five exceptions, every finite regular permutation group occurs as the automorphism group of a digraph.One of the corollaries: given a finite groupG of ordern, there is a commutative semigroupS of order 2n+2 such that AutSG. The problem whether a latticeL of order Cn with AutLG exists (for some constantC), remains open.     

Riordan group involutions

We study involutions in the Riordan group, especially those with combinatorial meaning. We give a new determinantal criterion for a matrix to be a Riordan involution and examine several classes of examples. A complete characterization of involutions in the Appell subgroup is developed. In another direction we find several examples that generalize the RNA matrix but are of independent interest.     

The existence of Room squares

The authors give a condensed proof of the existence of Room squares for positive odd sides except 3 and 5. Some areas of current research on Room squares are also discussed.Aequationes Mathematicae launches a systematic program of expository papers. We will endeavour to publish at least one in every volume.     

Ordinary and strong density continuity of complex analytic functions

In the paper we prove that the complex analytic functions are (ordinarily) density continuous. This stays in contrast with the fact that even such a simple function asG:²²,G(x,y)=(x,y ³ ), is not density continuous [1]. We will also characterize those analytic functions which are strongly density continuous at the given pointa . From this we conclude that a complex analytic functionf is strongly density continuous if and only iff(z)=a+bz, wherea, b andb is either real or imaginary.     

Spin invariant theory for the symmetric group

We formulate a theory of invariants for the spin symmetric group in some suitable modules which involve the polynomial and exterior algebras. We solve the corresponding graded multiplicity problem in terms of specializations of the Schur Q-functions and a shifted q-hook formula. In addition, we provide a bijective proof for a formula of the principal specialization of the Schur Q-functions.     

The automorphism group of a split metacyclic 2-group

This paper finds the order, structure and presentation for the automorphism group of any split metacyclic 2-group. Received: 2 August 2006     

The automorphism group of a split metacyclic p-group

This paper finds the order, structure and presentation for the automorphism group of a split metacyclic p-group, where p is odd. Received: 8 February 2006     

A torsion-free abelian group of finite rank exists whose quotient group modulo the square subgroup is not a nil-group

The first example of a finite rank torsion-free abelian group A such that the quotient group of A modulo the square subgroup of A is not a nil-group is indicated (in both cases of associative and general rings). In particular, the answer to the question posed by A.E. Stratton and M.C. Webb in [18], Abelian groups, nil modulo a subgroup, need not have nil quotient group, Publ. Math. Debrecen. 27 (1980), 127–130, is given for finite rank torsion-free groups. A relationship between nontrivial p-pure subgroups of the additive group of p-adic integers and nontrivial ? [p^?1]-submodules of the field of p-adic numbers is investigated. In particular, a bijective correspondence between these structures is proven using only elementary methods.     

On automorphism groups of Cayley graphs

LetX _G,H denote the Cayley graph of a finite groupG with respect to a subsetH. It is well-known that its automorphism groupA(X_G,H) must contain the regular subgroupL _G corresponding to the set of left multiplications by elements ofG. This paper is concerned with minimizing the index [A(X_G,H)L_G] for givenG, in particular when this index is always greater than 1. IfG is abelian but not one of seven exceptional groups, then a Cayley graph ofG exists for which this index is at most 2. Nearly complete results for the generalized dicyclic groups are also obtained.     

A note on Cartan matrices for symmetric groups

Using generating functions, a very simple explicit formula for the determinants of the p-Cartan matrices of symmetric groups is given. Our method works also when p is a composite number.Received: 5 September 2001     

Twisted second homology groups of the automorphism group of a free group

In this paper, we compute the second homology groups of the automorphism group of a free group with coefficients in the abelianization of the free group and its dual group except for the 2-torsion part, using combinatorial group theory.     

The quadratic moment problem for the unit circle and unit disk

For the quadratic complex moment problem , we obtain necessary and sufficient conditions for the existence of representing measures supported in the unit circleT or in the closed unit disk . We explicitly construct all finitely atomic representing measures supported inT or which have the fewest atoms possible. For the quadratic -moment problem in which the moment matrixM(1) is positive and invertible, there exists an ellipse D such that the minimal (3-atomic) representing measures are supported in the complement of the interior region of . Finally, we apply these results to obtain information on the location of the zeros of certain cubic polynomials.Dedicated to the memory of Velaho D. Bowman-FialkowBoth authors were partially supported by research grants from the National Science Foundation. The second-named author was also partially supported by an award from the State of New York/UUP Professional Development and Quality of Working Life Committee.     

The automorphism group of a rigid affine variety

An irreducible algebraic variety X is rigid if it admits no nontrivial action of the additive group of the ground field. We prove that the automorphism group of a rigid affine variety contains a unique maximal torus . If the grading on the algebra of regular functions defined by the action of  is pointed, the group  is a finite extension of . As an application, we describe the automorphism group of a rigid trinomial affine hypersurface and find all isomorphisms between such hypersurfaces.     

On the isomorphism of commutative group algebras

LetG andH be finite abelian groups and letF be an arbitrary field. One fundamental problem is that of determining necessary and sufficient conditions for the isomorphism of the group algebrasFG andFH. No solution has appeared in the literature. Nevertheless by combining the results of Berman, Perlis and Walker, Cohen, and Deskins and providing connecting arguments a complete solution can be obtained. It is the purpose of this note to present such a solution.     

Relatively pseudocomplemented semilattices amalgamate strongly

Letk be any finite or infinite cardinal andS _ω the symmetric group of denumerable infinite degree. It is shown that fori<k ifG _i is thei-th row of a matrix whose columns are allk-termed sequences of elements ofS _ω in each of which all but a finite number of terms are equal to the identity ofS _ω thenG _i 's (withG _i ⁻¹ 's defined in an obvious way and with coordinatewise multiplication amongG _i 's andG _i ⁻¹'s) generate the Free Group onk free generatorsG _i . Analogously, Free Abelian and other types of free groups are also constructed. Presented by L. Fuchs.