arXiv subjects

Piotr Grzeszczuk

Publications and source records attributed to Piotr Grzeszczuk.

5 recordsLinked to original sources

arXiv2026

Colouring bijections of finite $3$-groups

Let $G$ be a finite group. A bijection $σ\colon G\to G$ is a \emph{colouring bijection} if the three maps \[ Δ^{+}\colon x\mapsto x\,σ(x), \qquad Δ^{-}\colon x\mapsto x^{-1}σ(x), \qquad Δ^{c}\colon x\mapsto σ(x)^{-1}x\,σ(x) \] are again bijections of $G$. The first two conditions say that $σ$ is a strong complete mapping. The third is a genuinely nonabelian requirement. Our main theorem is that every noncyclic $3$-group not isomorphic to the modular group $M_{3^{r}}$ $(r\ge4)$ admits a colouring bijection. This is the exact analogue, for colouring bijections, of the theorem of Akhtar and Gagola on strong complete mappings. The notion has three equivalent readings. A colouring bijection properly colours the Cayley graph $\mathscr{G}_3(G)=Cay(G^{3},\mathbf S_3)$ with $|G|$ colours. It determines a triple of mutually orthogonal translation Latin squares based on $G$. In the orthomorphism graph of $G$ that triple is a triangle through the Cayley table. It also determines a common transversal of three arrays attached to $G$. These are the multiplication table, the division table, and the operation table of the conjugation quandle. The last of them is not a Latin square. Two consequences follow. Every noncyclic $3$-group $G\not\cong M_{3^{r}}$ $(r\ge4)$ carries three mutually orthogonal Latin squares of order $|G|$ based on $G$. Moreover $χ(\mathscr{G}_3(G))=|G|$ for every such group.

math.CO↗

arXiv2020

Locally nilpotent skew extensions of rings

We extend existing results on locally nilpotent differential polynomial rings to skew extensions of rings. We prove that if $\mathscr{G}=\{σ_t\}_{t\in T}$ is a locally finite family of automorphisms of an algebra $R$, $\mathscr{D}=\{δ_t\}_{t\in T}$ is a family of skew derivations of $R$ such that the prime radical $P$ of $R$ is strongly invariant under $\mathscr{D}$, then the ideal $P\langle T,\mathscr{G},\mathscr{D}\rangle^*$ of $R\langle T,\mathscr{G},\mathscr{D}\rangle$, generated by $P$, is locally nilpotent. We then apply this result to algebras with locally nilpotent derivations. We prove that any algebra $R$ over a field of characteristic $0$, having a surjective locally nilpotent derivation $d$ with commutative kernel, and such that $R$ is generated by $\ker d^2$, has a locally nilpotent Jacobson radical.

math.RA↗

arXiv2020

On the semiprimitivity of free skew extensions of rings

Let $X$ be a set of noncommuting variables of cardinality $card(X)\geqslant 2$, and ${\mathscr G}=\{σ_x\}_{x\in X}$, ${\mathscr D}=\{δ_x\}_{x\in X}$ be families of automorphisms and skew derivations of the ring $R$. It is proved that if the ring $R$ is semiprime Goldie, then the free skew extension $R[X;{\mathscr G},{\mathscr D}]$ is semiprimitive.

math.RA↗

arXiv2019

On the generic family of Cayley graphs of a finite group

Let $G$ be a finite group. For each $m>1$ we define the symmetric canonical subset $S=S(m)$ of the Cartesian power $G^m$ and we consider the family of Cayley graphs $\mathscr{G}_m(G)=Cay(G^m,S)$. We describe properties of these graphs and show that for a fixed $m>1$ and groups $G$ and $H$ the graphs $\mathscr{G}_m(G)$ and $\mathscr{G}_m(H)$ are isomorphic if and only if the groups $G$ and $H$ are isomorphic. We describe also the groups of automorphisms $\mathbf{Aut}(\mathscr{G}_m(G))$. It is shown that if $G$ is a non-abelian group, then $\mathbf{Aut}(\mathscr{G}_m(G))\simeq \big(G^m \rtimes \mathbf{Aut}(G)\big)\rtimes D_{m+1}$, where $D_{m+1}$ is the dihedral group of order $2m+2$. If $G$ is an abelian group (with some exceptions for $m=3$), then $\mathbf{Aut}(\mathscr{G}_m(G))\simeq G^m\rtimes \big(\mathbf{Aut}(G)\times S_{m+1}\big)$, where $S_{m+1}$ is the symmetric group of degree $m+1$. As an example of application we discuss relations between Cayley graphs $\mathscr{G}_m(G)$ and Bergman-Isaacs Theorem on rings with fixed-point-free group actions.

math.CO↗

arXiv2019

Generic identities for finite group actions

Let $G$ be a finite group of order $n$, and $Z_G=\mathbb{Z}\langleζ_{i,g}\mid g\in G,\ i=1,2,\dots,n\rangle$ be the free generic algebra, with canonical action of $G$ according to $(ζ_{i,g})^x=ζ_{i,x^{-1}g}$. It is proved that there exists a positive integer $\upsilon(G)$ such that for any $g_1,g_2,\dots, g_{n}\in G$ $$ \upsilon(G)\cdot ζ_{1,g_1}ζ_{2,g_2}\dotsζ_{n,g_{n}}=\sum_{i=1}^N γ_i a_i\mathbf{tr}_G(b_i)c_i, $$ where $γ_1,γ_2,\dots,γ_N$ are integers, and $a_i, b_i, c_i$ are monomials in $ζ_{i,g}$ such that ${\rm deg}(b_i)>0$ and ${\rm deg}(a_i)+{\rm deg}(b_i)+{\rm deg}(c_i)=n$. As a consequence, if $R$ is a ring (not necessarily unital) acted on by $G$, then the product $\upsilon(G)\cdot R^{n}$ is contained in the ideal $\langle\mathbf{tr}_G(R)\rangle$ generated by all traces $\mathbf{tr}_G(r)=\sum\limits_{g\in G}r^g$, $r\in R$. This gives the best possible nilpotence bound in Bergman-Isaacs theorem for finite group actions on non-commutative rings, which was a long standing problem. The main result was obtained by transferring the problem to certain family of Cayley graphs, and estimating their minimal eigenvalues by the clique numbers. It is proved that the clique number $ω(Γ)$ of any $k$-regular graph $Γ$ admits the Delsarte upper bound $ω(Γ)\leqslant\lfloor1-k/λ_{\rm min}\rfloor$.

math.RA↗