arXiv subjects
Publications and source records attributed to Mikhail Muzychuk.
We explicitly determine all CI-groups with respect to ternary relational structures that have the form $C \times D$, where $C$ is cyclic and $D$ is either a dicyclic group whose order is not divisible by $3$ or a dihedral group. Such groups are also CI-groups with respect to graphs and digraphs.
In this paper, we construct directed strongly regular graphs and divisible design graphs with new parameters merging some basic relations of so-called Tatra associations schemes. We also study the above association schemes, their fusions and isomorphisms.
Jordan schemes generalize association schemes in a similar way as Jordan algebras generalize the associative ones. It is well-known that association schemes of maximal rank are in one-to-one correspondence with groups (so-called thin schemes). In this paper, we classify Jordan schemes of maximal rank-to-order ratio and show that regular Jordan schemes correspond to a special class of Moufang loops, known as Ring Alternative loops.
For every odd prime power $q$, a family of pairwise nonisomorphic normal arc-transitive divisible design Cayley digraphs with isomorphic neighborhood designs over a Heisenberg group of order $q^3$ is constructed. It is proved that these digraphs are not distinguished by the Weisfeiler-Leman algorithm and have the Weisfeiler-Leman dimension $3$.
In the present paper, we study relative difference sets (RDSs) and linked systems of them. It is shown that a closed linked system of RDSs is always graded by a group. Based on this result, we also define a product of RDS linked systems sharing the same grading group. Further, we generalize the Davis-Polhill-Smith construction of a linked system of RDSs. Finally, we construct new linked system of RDSs in a Heisenberg group over a finite field and family of RDSs in an extraspecial $p$-group of exponent $p^2$. All constructions of new RDSs and their linked systems are based essentially on a usage of cyclotomic Schur rings.
This paper is a follow-up to (arXiv:2203.03687), in which the first author studied primitive association schemes lying between a tensor power $\mathcal{T}_m^d$ of the trivial association scheme and the Hamming scheme $\mathcal{H}(m,d)$. A question which arose naturally in that study was whether all primitive fusions of $\mathcal{T}_m^d$ lie between $\mathcal{T}_{m^e}^{d/e}$ and $\mathcal{H}(m^d, d/e)$ for some $e \mid d$. This note answers this question positively provided that $m$ is large enough. We similarly classify primitive fusions of the $d$th tensor power of a Johnson scheme on $\binom{m}{k}$ points provided $m$ is large enough in terms of $k$ and $d$.
We prove that the direct product of two coprime order elementary abelian groups of rank two, as well as the direct product of a cyclic group of prime order and a cyclic group of square free order are DCI-groups. The latter is a generalization of Muzychuk's result on cyclic groups (J. Combin. Theory Ser. A, 1995).
In 2003 Peter Cameron introduced the concept of a Jordan scheme and asked whether there exist Jordan schemes which are not symmetrisations of coherent configurations (proper Jordan schemes). The question was answered affirmatively by the authors last year and some of the examples were presented in an essay uploaded to the arXiv. In this paper we describe several infinite series of proper Jordan schemes and present first developments in the theory of Jordan schemes - a new class of algebraic-combinatorial objects.
Two matrix vector spaces $V,W\subset \mathbb C^{n\times n}$ are said to be equivalent if $SVR=W$ for some nonsingular $S$ and $R$. These spaces are congruent if $R=S^T$. We prove that if all matrices in $V$ and $W$ are symmetric, or all matrices in $V$ and $W$ are skew-symmetric, then $V$ and $W$ are congruent if and only if they are equivalent. Let $F: U\times\dots\times U\to V$ and $G: U'\times\dots\times U'\to V'$ be symmetric or skew-symmetric $k$-linear maps over $\mathbb C$. If there exists a set of linear bijections $φ_1,\dots,φ_k:U\to U'$ and $ψ:V\to V'$ that transforms $F$ to $G$, then there exists such a set with $φ_1=\dots=φ_k$.
In this paper, we find a strong new restriction on the structure of CI-groups. We show that, if $R$ is a generalised dihedral group and if $R$ is a CI-group, then for every odd prime $p$ the Sylow $p$-subgroup of $R$ has order $p$, or $9$. Consequently, any CI-group with quotient a generalised dihedral group has the same restriction, that for every odd prime $p$ the Sylow $p$-subgroup of the group has order $p$, or $9$. We also give a counter example to the conjecture that every BCI-group is a CI-group.
A special class of Jordan algebras over a field $F$ of characteristic zero is considered. Such an algebra consists of an $r$-dimensional subspace of the vector space of all square matrices of a fixed order $n$ over $F$. It contains the identity matrix, the all-one matrix; it is closed with respect to \correction{matrix transposition}, Schur-Hadamard (entrywise) multiplication and the Jordan product $A*B=\frac 12 (AB+BA)$, where $AB$ is the usual matrix product. The suggested axiomatics (with some natural additional requirements) implies an equivalent reformulation in terms of symmetric binary relations on a vertex set of cardinality $n$. The appearing graph-theoretical structure is called a Jordan scheme of order $n$ and rank $r$. A significant source of Jordan schemes stems from the symmetrization of association schemes. Each such structure is called a non-proper Jordan scheme. The question about the existence of proper Jordan schemes was posed a few times by Peter J. Cameron. In the current text an affirmative answer to this question is given. The first small examples presented here have orders $n=15,24,40$. Infinite classes of proper Jordan schemes of rank 5 and larger are introduced. A prolific construction for schemes of rank 5 and order $n=\binom{3^d+1}{2}$, $d\in {\mathbb N}$, is outlined. The text is written in the style of an essay. The long exposition relies on initial computer experiments, a large amount of diagrams, and finally is supported by a number of patterns of general theoretical reasonings. The essay contains also a historical survey and an extensive bibliography.
For every pair of distinct primes $p$, $q$ we prove that $\mathbb{Z}_p^3 \times \mathbb{Z}_q$ is a CI-group with respect to binary relational structures.
Let ${\frak K}$ be a class of combinatorial objects invariant with respect to a given regular cyclic group. It is proved that the isomorphism of any two objects $X,Y\in{\frak K}$ can be tested in polynomial time in sizes of $X$ and $Y$.
Given a finite group $G$ and two unitary $G$-representations $V$ and $W$, possible restrictions on Brouwer degrees of equivariant maps between representation spheres $S(V)$ and $S(W)$ are usually expressed in a form of congruences modulo the greatest common divisor of lengths of orbits in $S(V)$ (denoted $α(V)$). Effective applications of these congruences is limited by answers to the following questions: (i) under which conditions, is $α(V)>1$? and (ii) does there exist an equivariant map with the degree easy to calculate? In the present paper, we address both questions. We show that $α(V)>1$ for each irreducible non-trivial $C[G]$-module if and only if $G$ is solvable. For non-solvable groups, we use 2-transitive actions to construct complex representations with non-trivial $α$-characteristic. Regarding the second question, we suggest a class of Norton algebras without 2-nilpotents giving rise to equivariant quadratic maps, which admit an explicit formula for the Brouwer degree.
A cycle base of a permutation group is defined to be a maximal set of its pairwise non-conjugate regular cyclic subgroups. It is proved that a cycle base of a permutation group of degree $n$ can be constructed in polynomial time in~$n$.
Given a prime $p$, we construct a permutation group containing at least $p^{p-2}$ non-conjugated regular elementary abelian subgroups of order $p^3$. This gives the first example of a permutation group with exponentially many non-conjugated regular subgroups.
In this paper we study finite groups which have Cayley isomorphism property with respect to Cayley maps, CIM-groups for a brief. We show that the structure of the CIM-groups is very restricted. It is described in Theorem~\ref{111015a} where a short list of possible candidates for CIM-groups is given. Theorem~\ref{111015c} provides concrete examples of infinite series of CIM-groups.
In this paper it is shown that every finite cyclic group satisfies the CI-property for the class of balanced configurations.