Regular Automorphism Research Articles

Tournaments with prescribed regular automorphism group

A tournamentX is a TRR for a groupG if (a)G acts regularly on the vertices ofX and (b) Aut(X) is isomorphic toG. We correct some previous work of Babai and Imrich by showing thatZ23 andZ33 are the only groups of odd order without TRR's. Our methods are perhaps of independent interest, since we use a probabilistic approach.

The 2-adic affine building of type [formula omitted] and its finite projections

A certain “free” group U is constructed that is generated by three elements of order 3 which pairwise generate a Frobenius group of order 21 and it is shown that U operates regularly on the affine building of type A ̃ 2 over the field of 2-adic numbers. As a result an infinite series of finite rank 3 geometries is obtained whose rank 2 residues are projective planes of order 2, and which possess a regular automorphism group isomorphic to SL 3( p) or SU 3( p) for some prime p.

A note on planar difference sets

It is proven that if G is an abelian regular automorphism group of a projective plane of order n and if p is a prime dividing n exactly once, then a certain identity holds in the group algebra F p|G|. As a corollary we obtain that n = p if p = 2 or 3.

Finite groups admitting an almost regular automorphism of order 3

Finite groups admitting an almost regular automorphism of order 3

On the Automorphism Groups of almost all Cayley Graphs

In spite of the difficulties which arise in constructing Cayley graphs with given regular automorphism group G , we conjecture that, unless G belongs to a known class of exceptions, almost all Cayley graphs of G have G for their full automorphism group . Here we prove this conjecture for nilpotent groups of odd order .

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 exceptZ3 ×Z3 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 byZ2n+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.

Infinite quasigroups with given regular automorphism groups

Infinite quasigroups with given regular automorphism groups

A solvable group admitting a regular splitting automorphism of prime order is nilpotent

A solvable group admitting a regular splitting automorphism of prime order is nilpotent

Infinite digraphs with given regular automorphism groups

We prove that, given an infinite group G there is a directed graph X such that its automorphism group A(X) is the regular representation of G . The proof uses combinatorial set theoretic arguments (Erdös-Rado partition calculus). One of the lemmas asserts that, if | G |>2 κ then G contains a subset H of power κ + satisfying x −1 y=y −1 z⇒x=z (x, y, z ∈ H) Corollary . Given an infinite group G there exist a simple graph X a partially ordered set P , and a commutative semigroup S such that A(X)≅A(P)≅A(S)≅G . Moreover A(X) and A(P) each have three orbits, while A(S) has five orbits.

On the Commutativity of Semi-Simple Associative Algebras

Recall that an automorphism ϕ of an algebra A is called regular if, for all x∊A, ϕ(x) = x implies x = 0. For various types of algebras it is known that the existence of a regular automorphism implies that the algebra is radical. Here, for associative algebras, we relax the restrictions on the fixed points so that all automorphisms of commutative algebras immediately satisfy the hypothesis on the fixed points.

Block designs admitting regular automorphisms

Block designs admitting regular automorphisms

Abelian and metabelian groups with regular automorphisms and semiautomorphisms

In this paper we describe all abelian groups which can be expanded as the direct product of cyclic subgroups which have regular automorphisms of given prime order and also find the necessary and sufficient conditions for the existence of regular semiautomorphisms of prime order for metabelian groups the orders of the elements of which are bounded as a set.

An example of an automorphism which is a mixing of all degrees but which is not a Kolmogorov automorphism

Rohlin (7) has shown that Kolmogorov automorphisms are mixings of all degrees. Later Sucheston(9) introduced regular automorphisms and after showing that they are mixings of all degrees asks if the converse of either of the above theorems holds. Since the equivalence of Kolmogorov and regular automorphisms follows from (1) and (8) we have only one problem. We show that the converse of either of the above theorems is false by constructing a stationary Gaussian process which is a mixing of all degrees but which is not a Kolmogorov automorphism.

Groups with regular automorphisms of order four

Groups with regular automorphisms of order four

Note on Non-Associative Rings with Regular Automorphisms

Note on Non-Associative Rings with Regular Automorphisms