arXiv subjects

Czesław Bagiński

Publications and source records attributed to Czesław Bagiński.

3 recordsLinked to original sources

arXiv2024

The Modular Isomorphism Problem -- the alternative perspective on counterexamples

As a result of impressive research arXiv:2106.07231, D. Garc\'ıa-Lucas, Á. del Río and L. Margolis defined an infinite series of non-isomorphic $2$-groups $G$ and $H$, whose group algebras $\mathbb{F}G$ and $\mathbb{F}H$ over the field $\mathbb{F}=\mathbb{F}_2$ are isomorphic, solving negatively the long-standing Modular Isomorphism Problem (MIP). In this note we give a different perspective on their examples and show that they are special cases of a more general construction. We also show that this type of construction for $p>2$ does not provide a similar counterexample to the MIP.

math.GR↗

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↗

arXiv2006

The modular isomorphism problem for finite $p$-groups with a cyclic subgroup of index $p^2$

Let $p$ be a prime number, $G$ be a finite $p$-group and $K$ be a field of characteristic $p$. The Modular Isomorphism Problem (MIP) asks whether the group algebra $KG$ determines the group $G$. Dealing with MIP, we investigated a question whether the nilpotency class of a finite $p$-group is determined by its modular group algebra over the field of $p$ elements. We give a positive answer to this question provided one of the following conditions holds: (i) $\exp G=p$; (ii) $\cl(G)=2$; (iii) $G'$ is cyclic; (iv) $G$ is a group of maximal class and contains an abelian subgroup of index $p$.

math.RA↗