Abstract
t. Let G be a transitive permutation group on a finite set Ω and recall that a
base for G is a subset of Ω with trivial pointwise stabiliser. The base size of G, denoted b(G), is the minimal size of a base. If b(G) = 2 then we can study the Saxl graph Σ(G) of G, which has vertex set Ω and two vertices are adjacent if and only if they form a base. This is a vertex-transitive graph, which is conjectured to be connected with diameter at most 2 when G is primitive. In this paper, we combine probabilistic and computational methods to prove a strong form of this conjecture for all almost simple primitive groups with soluble point stabilisers. In this setting, we also establish best possible lower bounds on the clique and independence numbers of Σ(G) and we determine the groups with a unique regular suborbit, which can be interpreted in terms of the valency of Σ(G).
base for G is a subset of Ω with trivial pointwise stabiliser. The base size of G, denoted b(G), is the minimal size of a base. If b(G) = 2 then we can study the Saxl graph Σ(G) of G, which has vertex set Ω and two vertices are adjacent if and only if they form a base. This is a vertex-transitive graph, which is conjectured to be connected with diameter at most 2 when G is primitive. In this paper, we combine probabilistic and computational methods to prove a strong form of this conjecture for all almost simple primitive groups with soluble point stabilisers. In this setting, we also establish best possible lower bounds on the clique and independence numbers of Σ(G) and we determine the groups with a unique regular suborbit, which can be interpreted in terms of the valency of Σ(G).
| Original language | English |
|---|---|
| Pages (from-to) | 1053-1087 |
| Number of pages | 35 |
| Journal | Algebraic Combinatorics |
| Volume | 5 |
| Issue number | 5 |
| Early online date | 9 Nov 2022 |
| DOIs | |
| Publication status | E-pub ahead of print - 9 Nov 2022 |
Bibliographical note
Funding Information:Acknowledgements. Both authors thank Eamonn O’Brien for his assistance with several computations in this paper. The second author thanks the China Scholarship Council for supporting his doctoral studies at the University of Bristol and he thanks the Southern University of Science and Technology (SUSTech) for their generous hospitality during a visit in 2021.
Publisher Copyright:
© The author(s), 2022.