On Sylow Subgraphs of Vertex-Transitive Self-Complementary Graphs

One of the basic facts of group theory is that each finite groupcontains a Sylow p-subgroup for each prime p which divides theorder of the group. In this note we show that each vertex-transitiveself- complementary graph has an analogous property. As a consequenceof this fact, we obtain that each prime divisor p of the orderof a vertex-transitive self-complementary graph satisfies thecongruence p^m 1(mod 4), where p^m is the highest power of pwhich divides the order of the graph. 1991 Mathematics SubjectClassification 05C25, 20B25.     

On groups with conjugacy classes of distinct sizes

A finite group G is called an ah-group if any two distinct conjugacy classes of G have distinct cardinality. We show that if G is an ah-group, then the non-abelian socle of G is isomorphic to one of the following:

1. , for 1a5, a≠2.

2. A₈.

3. PSL(3,4)^e, for 1e10.

4. A₅×PSL(3,4)^e, for 1e10.

Based on this result, we virtually show that if G is an ah-group with π(G) 2,3,5,7 , then F(G)≠1, or equivalently, that G has an abelian normal subgroup.In addition, we show that if G is an ah-group of minimal size which is not isomorphic to S₃, then the non-abelian socle of G is either trivial or isomorphic to one of the following:

1. , for 3a5.

2. PSL(3,4)^e, for 1e10.

Our research lead us to interesting results related to transitivity and homogeneousity in permutation groups, and to subgroups of wreath products of form Z₂S_n. These results are of independent interest and are located in appendices for greater autonomy.     

On association schemes all elements of which have valency 1 or 2

In the present note, we investigate schemes S in which each element s satisfies n_s2 and n_s^*s≠2. We show that such a scheme is schurian. More precisely, we show that it is isomorphic to G//t, where G is a finite group and t an involution of G weakly closed in C_G(t).

Groups G with an involution t weakly closed in C_G(t) have been described in Glauberman's Z^*-Theorem [G. Glauberman, Central elements in core-free groups, J. Algebra 4 (1966) 403–420] with the help of the largest normal subgroup of G having odd order.     

Tensor Rank: Matching Polynomials and Schur Rings

We study the polynomial equations vanishing on tensors of a given rank. By means of polarization we reduce them to elements  $A$ of the group algebra ${\mathbb {Q}}[S_n\times S_n]$ and describe explicit linear equations on the coefficients of  $A$ to vanish on tensors of a given rank. Further, we reduce the study to the Schur ring over the group $S_n\times S_n$ that arises from the diagonal conjugacy action of  $S_n$ . More closely, we consider elements of ${\mathbb {Q}}[S_n\times S_n]$ vanishing on tensors of rank $n-1$ and describe them in terms of triples of Young diagrams, their irreducible characters, and nonvanishing of their Kronecker coefficients. Also, we construct a family of elements in ${\mathbb {Q}}[S_n\times S_n]$ vanishing on tensors of rank $n-1$ and illustrate our approach by a sharp lower bound on the border rank of an explicitly produced tensor. Finally, we apply this construction to prove a lower bound $5n^2/4$ on the border rank of the matrix multiplication tensor (being, of course, weaker than the best known one $(2-\epsilon )\cdot n^2$ , due to Landsberg, Ottaviani).     

Probability characteristics of brownian motion in a stochastic potential profile of a specific type

Some probability characteristics of the Brownian motion in a symmetric potential profile with two equilibrium states subjected to a random force are obtained. Two types of potential fluctuations are considered: the delta-correlated Gaussian noise and the stochastic telegraph process with Poisson statistics of jumps. The stationary probability distributions of the particle coordinate are found, and the dependence on the properties of parametric and additive noise is studied. It is shown that nonzero equilibrium states approach each other and vanish as a result of strong potential fluctuations. Relaxation of intensity and variance of coordinate fluctuations are studied numerically for the case of delta-correlated random forces. The influence of the value of parametric and additive noise, system nonlinearity, and initial conditions on the relaxation process is determined. Architectural and Civil Engineering University, Nizhny Novgorod, Russia. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 41, No. 10, pp. 1290–1300, October 1998.     

A Solution of a Problem of A. E. Brouwer

Let q=p^e 1(mod 4) be a prime power, and let ^(q) be the Paley graph over the finite field . Denote by ^(q) the subgraph of ^(q) induced on the set of non-zero squares of . In this paper the full automorphism group of ^(q) is determined affirming the conjecture of Brouwer [Des. Codes Cryptograph. 21, 69–76 (2000)]. The proof combines spectral and Schur ring techniques.