Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Quasiprimitive and bi-quasiprimitive highly-arc-transitive digraphs and finite simple groups

Lei Chen, Cheryl Praeger

TL;DR

This work extends the notion of an $H$-normal quotient to $H$-subnormal quotients and uses bipartite halves to connect finite, vertex-transitive, highly arc-transitive digraphs with vertex-quasiprimitive almost-simple automorphism groups. It proves that for large $s$ there is a dichotomy: either a $G$-normal quotient is a directed cycle, or there exists an $(L,t)$-arc-transitive digraph with $L$ almost simple and vertex-quasiprimitive, where $t$ is roughly half of $s$, tying large-arc-transitivity to almost-simple vertex actions. The authors construct infinite families of examples with alternating and symmetric groups and introduce a new bipartite-doubling construction that yields vertex-bi-quasiprimitivity for arbitrarily large $s$, alongside a suite of open problems and reductions between quasiprimitive, bi-quasiprimitive, and diagonal-type cases. Together, these results advance understanding of how group-theoretic action types constrain and generate highly arc-transitive digraphs, with potential implications for classifying finite arc-transitive structures.

Abstract

We extend the notion of an $H$-normal quotient digraph of an $H$-vertex-transitive digraph to that of an $H$-subnormal quotient digraph. Using these concepts, together with bipartite halves of bipartite digraphs, we show that, for each finite connected $H$-vertex-transitive, $(H,s)$-arc-transitive digraph with $s\geqslant6$, either some $H$-normal quotient is a directed cycle of length at least $3$, or there is an $(L,t)$-arc-transitive digraph with $t\geqslant (s-3)/2$, and $L$ a vertex-quasiprimitive almost simple group with socle a composition factor of $H$. This connection demonstrates that, to understand finite $s$-arc-transitive digraphs with large $s$, those admitting a vertex-quasiprimitive almost simple $s$-arc-transitive subgroup of automorphisms play a central role. We show that for each $s$ and each odd valency $k$, there are infinitely many $(H,s)$-arc-transitive digraphs of valency $k$ with $H$ a finite alternating group. In addition we discovered a novel construction which takes as input a connected non-bipartite $H$-vertex-transitive, $(H,s)$-arc-transitive digraph, and outputs a connected bipartite $G$-vertex-transitive, $(G,2s)$-arc-transitive digraph with $G=(H\times H).2$. This leads to construction of vertex-bi-quasiprimitive $s$-arc-transitive digraphs, for arbitrarily large $s$. Our investigations yield several new open problems.

Quasiprimitive and bi-quasiprimitive highly-arc-transitive digraphs and finite simple groups

TL;DR

This work extends the notion of an -normal quotient to -subnormal quotients and uses bipartite halves to connect finite, vertex-transitive, highly arc-transitive digraphs with vertex-quasiprimitive almost-simple automorphism groups. It proves that for large there is a dichotomy: either a -normal quotient is a directed cycle, or there exists an -arc-transitive digraph with almost simple and vertex-quasiprimitive, where is roughly half of , tying large-arc-transitivity to almost-simple vertex actions. The authors construct infinite families of examples with alternating and symmetric groups and introduce a new bipartite-doubling construction that yields vertex-bi-quasiprimitivity for arbitrarily large , alongside a suite of open problems and reductions between quasiprimitive, bi-quasiprimitive, and diagonal-type cases. Together, these results advance understanding of how group-theoretic action types constrain and generate highly arc-transitive digraphs, with potential implications for classifying finite arc-transitive structures.

Abstract

We extend the notion of an -normal quotient digraph of an -vertex-transitive digraph to that of an -subnormal quotient digraph. Using these concepts, together with bipartite halves of bipartite digraphs, we show that, for each finite connected -vertex-transitive, -arc-transitive digraph with , either some -normal quotient is a directed cycle of length at least , or there is an -arc-transitive digraph with , and a vertex-quasiprimitive almost simple group with socle a composition factor of . This connection demonstrates that, to understand finite -arc-transitive digraphs with large , those admitting a vertex-quasiprimitive almost simple -arc-transitive subgroup of automorphisms play a central role. We show that for each and each odd valency , there are infinitely many -arc-transitive digraphs of valency with a finite alternating group. In addition we discovered a novel construction which takes as input a connected non-bipartite -vertex-transitive, -arc-transitive digraph, and outputs a connected bipartite -vertex-transitive, -arc-transitive digraph with . This leads to construction of vertex-bi-quasiprimitive -arc-transitive digraphs, for arbitrarily large . Our investigations yield several new open problems.

Paper Structure

This paper contains 15 sections, 14 theorems, 13 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Commentary and open problems
  3. The simple groups arising in Theorem \ref{['t:main']}.
  4. Vertex-quasiprimitive $s$-arc-transitive graphs of PA type
  5. Vertex-quasiprimitive $s$-arc-transitive digraphs of diagonal type (SD and CD)
  6. Vertex-bi-quasiprimitive $s$-arc-transitive digraphs
  7. Basic $s$-arc-transitive digraphs of cycle type
  8. Reduction steps for $(G,s)$-arc-transitive digraphs
  9. Normal and subnormal quotients
  10. Bipartite halves
  11. Quasiprimitive digraphs
  12. Bi-quasiprimitive digraphs
  13. Construction of bi-quasiprimitive highly-arc-transitive digraphs
  14. Proof of Theorem \ref{['t:main']}
  15. Quasiprimitive and bi-quasiprimitive digraphs admitting $\mathrm{Alt}(n)$ or $\mathrm{Sym}(n)$

Key Result

Theorem 1.2

Let $(\Gamma^{(1)},G^{(1)},s^{(1)})\in{\mathscr{D}}$ with $s^{(1)}$ as in Table tab:main. Then there is a reduction sequence as in Definition d:reln with each $(\Gamma^{(i)},G^{(i)},s^{(i)})\in{\mathscr{D}}$, such that $n, s^{(n)}$ are as in Table tab:main, and $G^{(n)}$ is almost simple and vertex-quasiprimitive on $\Gamma^{(n)}$ with socle a composition factor of $G^{(1)}$.

Theorems & Definitions (32)

  • Definition 1.1
  • Theorem 1.2
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Definition 2.4
  • Proposition 2.5
  • proof
  • ...and 22 more