Regular Automorphism Research Articles

Birational automorphisms of a three-dimensional double quadric with an elementary singularity

It is proved that the group of birational automorphisms of a three-dimensional double quadric with a singular point arising from a double point on the branch divisor is a semidirect product of the free group generated by birational involutions of a special form and the group of regular automorphisms. The proof is based on the method of 'untwisting' maximal singularities of linear systems.

Cocyclic Generalised Hadamard Matrices and Central RelativeDifference Sets

Cocyclic matrices have the form M = [ \psi(g, h)]_{g, h \in G}, where G is a finite group, C is a finite abelian group and \psi : G \times G \rightarrow C is a (two-dimensional) cocycle; that is, \psi(g, h) \psi(gh, k) = \psi(g, hk) \psi(h, k), \forall g, h, k \in G. This expression of the cocycle equation for finite groups as a square matrix allows us to link group cohomology, divisible designs with regular automorphism groups and relative difference sets. Let G have order v and C have order w, with w|v. We show that the existence of a G-cocyclic generalised Hadamard matrix GH (w, v/w) with entries in C is equivalent to the existence of a relative ( v, w, v, v/w)-difference set in a central extension E of C by G relative to the central subgroupC and, consequently, is equivalent to the existence of a (square) divisible ( v, w, v, v/w)-design, class regular with respect to C, with a central extension E of C as regular group of automorphisms. This provides a new technique for the construction of semiregular relative difference sets and transversal designs, and generalises several known results.

Analogue of G. Higman's Theorem for Commutative Torsion-Free Rings

In this paper we consider finite rank torsion-free rings, which have almost regular automorphisms of prime order (a non-trivial automorphism is called almost regular if it has only trivial fixed points, i.e. zero and the elements of a ring linear dependent on its identity). The main result of this paper is the analogue of G. Higman's known Theorem [1] on almost regular automorphism for commutative finite rank torsion-free rings.

Almost regular automorphisms of finite groups of bounded rank

Almost regular automorphisms of finite groups of bounded rank

New constructions of group divisible designs

We introduce new difference methods and apply them to construct group divisible designs with regular automorphism groups.

On circulant digraphs with regular automorphism groups

We prove that the cyclic groupZ n (n ≥ 3) has ak-regular digraph regular representation if and only if 0 <k <n − 1.

Nilpotent groups admitting an almost regular automorphism of order four

We consider locally nilpotent periodic groups admitting an almost regular automorphism of order 4. The following are results are proved: (1) If a locally nilpotent periodic group G admits an automorphism ϕ of order 4 having exactly m<∞ fixed points, then (a) the subgroup {ie176-1} contains a subgroup of m-bounded index in {ie176-2} which is nilpotent of m-bounded class, and (b) the group G contains a subgroup V of m-bounded index such that the subgroup {ie176-3} is nilpotent of m-bounded class (Theorem 1); (2) If a locally nilpotent periodic group G admits an automorphism ϕ of order 4 having exactly m<∞ fixed points, then it contains a subgroup V of m-bounded index such that, for some m-bounded number f(m), the subgroup {ie176-4}, generated by all f(m) th powers of elements in {ie176-5} is nilpotent of class ≤3 (Theorem 2).

Regular automorphism groups on partial geometries

Recently, Ghinelli (Geom Dedicata (1992) 165–174) had studied the generalized quadrangles which admits automorphism groups acting regularly on the points. In this paper, we generalize her idea to partial geometries, pg( s + 1, t + 1, α). Some examples and basic properties are given. In particular, we prove that under certain conditions on the automorphism group and the lines, such a geometry is a translation net. Applying the results to the case when s = t and the automorphism group G is abelian, we find that either the geometry is a translation net or all the lines of the geometry are generated by a subset of G. Also, for this case, we conjecture that the parameter α is either s or s + 1, except ( s, α) = (5, 2), and we have checked that it is true for s ⩽ 500.

Torsion-free rings with almost regular automorphisms

In this paper we consider finite rank torsion-free rings, which have almost regular automorphisms (a non-trivial automorphism is calledalmost regular if it has only trivial fixed points, i.e. zero and the elements of a ring linear dependent on its identity). The main results of this paper are the characterization of some types of rings which have almost regular automorphism of the finite order (the analogue of known G.Higman’s Theorem[l]) and the characterization of all rings, all of whose non-trivial automorphisms are almost regular.

Fixed points of automorphisms of Lie rings and locally finite groups

Let P be a locally finite group of prime exponent p. We prove that if P admits a finite soluble automorphism group G of order n coprime to p, such that the fixed point group CP(G)is soluble of derived length d, then P is nilpotent of class bounded by a function of p, n, and d. A similar statement is shown to hold for Lie (p - 1)-Engel algebras; it is analogous to the Bergman-Isaacs theorem proved for associative rings, provided the condition of being soluble for an automorphism group is added. Our proof is based on a generalization of Kreknin's theorem concerning the solubility of Lie rings with a regular automorphism of finite order. This generalization, giving an affirmative answer to a question of Winter and extending one of his results to the case of infinitedimensional Lie algebras, is interesting in its own right. Moreover, we use a generalization of Higgins' theorem on the nilpotency of soluble Lie Engel algebras.

Some families of semibiplanes

A semibiplane is a connected finite incidence structure satisfying (i) every pair of points are incident with 0 or 2 blocks (ii) every pair of blocks are incident with 0 or 2 points. We give several constructions for combining semibiplanes to produce new ones. Some of these constructions are iterative and some require the input semibiplanes to have special properties (such as divisibility, regular automorphism groups, polarities). There are two well-known constructions which supply infinite families of semibiplanes with the properties we require: one based on involutions in projective planes, the other having point set and block set being the vectors of V n (2). We review these constructions and show how our methods may be used to construct several more infinite families of semibiplanes.

FINITE -GROUPS ADMITTING -AUTOMORPHISMS WITH FEW FIXED POINTS

The following theorem is proved: if a finite -group admits an automorphism of order having exactly fixed points, then it contains a subgroup of -bounded index that is solvable of -bounded derived length. The proof uses Kreknin's theorem stating that a Lie ring admitting a regular (that is, without nontrivial fixed points) automorphism of finite order , is solvable of -bounded derived length . Some techniques from the theory of powerful -groups are also used, especially, from a recent work of Shalev, who proved that, under the hypothesis of the theorem, the derived length of is bounded in terms of , , and . The following general proposition is also used (this proposition is proved on the basis of Kreknin's theorem with the help of the Mal'tsev correspondence, given by the Baker-Hausdorff formula): if a nilpotent group of class admits an automorphism of finite order , then, for some -bounded number , the derived subgroup is contained in the normal closure of the centralizer . The scheme of the proof of the theorem is as follows. Standard arguments show that may be assumed to be a powerful -group. Next, it is proved that is nilpotent of -bounded class. Then the proposition is applied to . There exist explicit upper bounds for the functions from the statement of the theorem. Bibliography: 22 titles.

CONNECTIVITY AND SPECTRUM IN A GRAPH WITH A REGULAR AUTOMORPHISM GROUP OF ODD ORDER

Let a finite group G of odd order n act regularly on a connected (multi-)graph Γ. That is, no group element other than the identity fixes any vertex. Then the “quotient graph” Δ under the action is the induced graph of orbits. We give a result about the connectivity of Γ and Δ in terms of their numbers of labeled spanning trees. In words, the spanning tree count of the graph is equal to n, the order of the given regular automorphism group, times the spanning tree count of the graph of orbits, times a perfect square integer. There is a dual result on the Laplacian spectrum saying that the multiset of Laplacian eigenvalues for the main graph is the disjoint union of the multiset for the quotient graph together with a multiset all of whose elements have even multiplicity. Specializing to the case of one orbit, we observe that a Cayley graph of odd order has spanning tree count equal to n times a square, and that that the Laplacian spectrum consists of the value 0 together with other doubled eigenvalues. These results are based on a study of matrices (and determinants) that consist of blocks of group-matrices. The generic determinant for such a matrix with the additional property of symmetry will have a dominanting square factor in its (multinomial) factorization. To show this, we make use of the Feit-Thompson theorem which provides a normal tower for an odd-order group, and perform a similarity conjugation with a fixed integer, unimodal matrix. Additional related results are given for certain group-matrices “twisted” by a group of automorphisms, generalizing the “g-circulants” of P.J. Davis.

On nilpotent groups having an almost regular automorphism of prime order

On nilpotent groups having an almost regular automorphism of prime order

Groups and Lie Algebras with Almost Regular Automorphisms

It is proved that if a locally nilpotent group G admits an almost regular automorphism of prime order p then G contains a nilpotent subgroup G1 such that |G : G1|≤ƒ(p, m) and the class of nilpotency of G1ƒg(p), where ƒ is a function on p and the number of fixed elements m and g depends on p only. An analog is proved for Lie rings (not necessarily locally nilpotent). These give an affirmative answer to the questions raised by Khukhro.

Lie algebras admitting a hypercentrally regular automorphism

Lie algebras admitting a hypercentrally regular automorphism

CORRECTION

The title of the paper by E. I. Khukhro (pp. 51-63) should read: Groups and Lie rings admitting an almost regular automorphism of prime order.

On almost regular automorphisms of prime order

On almost regular automorphisms of prime order

Subrings of invariants of a finite group of automorphisms of a Jordan ring

In the theory of associative rings various authors have investigated the connection between the rings R G and R. An important theorem in this direction was proved in [i]: If the ring R is iGl-torsion-free (G a finite group of automorphisms of R), then nilpotency of R G implies nilpotency of R. It was shown in [2] that, under the same conditions, if R G is a PI-ring, then R is a PI-ring. In nonassociative rings, mainly regular (i.e., having no fixed points) automorphisms of finite orders have been studied [3, 4]. We must mention the paper [5], in which were investigated the connections between properties of R G and R in the case where G is a finite group of Jordan automorphisms of an associative ring R.

On multiplier theorems of relative quotient sets

A group divisible design with regular automorphism group G can be described entirely in terms of G, a subgroup H of G, and a subset Δ of G called relative quotient set of G with respect to H. In this article we are interested in multipliers of relative quotient sets. We use methods due to Ott to generalize a multiplier theorem of Ko and Ray-Chaudhuri to not necessarily abelian groups.