On kernel isomorphisms of $m$-Cayley digraphs and finite $2$PCI-groups
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866918058913693696 |
|---|---|
| author | Zhang, Xing Feng, Yan-Quan Zhou, Jin-Xin Yin, Fu-Gang |
| author_facet | Zhang, Xing Feng, Yan-Quan Zhou, Jin-Xin Yin, Fu-Gang |
| contents | The isomorphism problem for digraphs is a fundamental problem in graph theory. In this paper, we consider this problem for $m$-Cayley digraphs which are generalization of Cayley digraphs. Let $m$ be a positive integer. A digraph admitting a group $G$ of automorphisms acting semiregularly on the vertices with exactly $m$ orbits is called an $m$-Cayley digraph of $G$. In our previous paper, we developed a theory for $m$-Cayley isomorphisms of $m$-Cayley digraphs, and classified finite $m$CI-groups for each $m\geq 2$, and finite $m$PCI-groups for each $m\geq 4$. The next natural step is to classify finite $m$PCI-groups for $m=2$ or $3$. Note that BCI-groups form an important subclass of the $2$PCI-groups, which were introduced in 2008 by Xu et al. Despite much effort having been made on the study of BCI-groups, the problem of classifying finite BCI-groups is still widely open.
In this paper, we prove that every finite $2$PCI-group is solvable, and its Sylow $3$-subgroup is isomorphic to $Z_3, Z_3\times Z_3$ or $Z_9$, and Sylow $p$-subgroup with $p\not=3$ is either elementary abelian, or isomorphic to $Z_4$ or $Q_8$. We also introduce the kernel isomorphisms of $m$-Cayley digraphs, and establish some useful theory for studying this kind of isomorphisms. Using the results of kernel isomorphisms of $m$-Cayley digraphs together with the results on $2$PCI-groups, we give a proper description of finite BCI-groups, and in particular, we obtain a complete classification of finite non-abelian BCI-groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_12306 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On kernel isomorphisms of $m$-Cayley digraphs and finite $2$PCI-groups Zhang, Xing Feng, Yan-Quan Zhou, Jin-Xin Yin, Fu-Gang Combinatorics 20B25, 05C25 The isomorphism problem for digraphs is a fundamental problem in graph theory. In this paper, we consider this problem for $m$-Cayley digraphs which are generalization of Cayley digraphs. Let $m$ be a positive integer. A digraph admitting a group $G$ of automorphisms acting semiregularly on the vertices with exactly $m$ orbits is called an $m$-Cayley digraph of $G$. In our previous paper, we developed a theory for $m$-Cayley isomorphisms of $m$-Cayley digraphs, and classified finite $m$CI-groups for each $m\geq 2$, and finite $m$PCI-groups for each $m\geq 4$. The next natural step is to classify finite $m$PCI-groups for $m=2$ or $3$. Note that BCI-groups form an important subclass of the $2$PCI-groups, which were introduced in 2008 by Xu et al. Despite much effort having been made on the study of BCI-groups, the problem of classifying finite BCI-groups is still widely open. In this paper, we prove that every finite $2$PCI-group is solvable, and its Sylow $3$-subgroup is isomorphic to $Z_3, Z_3\times Z_3$ or $Z_9$, and Sylow $p$-subgroup with $p\not=3$ is either elementary abelian, or isomorphic to $Z_4$ or $Q_8$. We also introduce the kernel isomorphisms of $m$-Cayley digraphs, and establish some useful theory for studying this kind of isomorphisms. Using the results of kernel isomorphisms of $m$-Cayley digraphs together with the results on $2$PCI-groups, we give a proper description of finite BCI-groups, and in particular, we obtain a complete classification of finite non-abelian BCI-groups. |
| title | On kernel isomorphisms of $m$-Cayley digraphs and finite $2$PCI-groups |
| topic | Combinatorics 20B25, 05C25 |
| url | https://arxiv.org/abs/2506.12306 |