Abstract
Let G be a finite primitive permutation group on a set with point stabiliser H. Recall that a subset of is a base for G if its pointwise stabiliser is trivial. We define the base size of G, denoted b(G, H), to be the minimal size of a base for G. Determining the base size of a group is a fundamental problem in permutation group theory, with a long history stretching back to the 19th century. Here one of our main motivations is a theorem of Seress from 1996, which states that b(G, H) 6 4 if G is soluble. In this paper we extend Seress’s result by proving that b(G, H) 6 5 for all finite primitive groups G with a soluble point stabiliser H. This bound is best possible. We also determine the exact base size for all almost
simple groups and we study random bases in this setting. For example, we prove that the probability that 4 random elements in form a base tends to 1 as |G| tends to infinity.
simple groups and we study random bases in this setting. For example, we prove that the probability that 4 random elements in form a base tends to 1 as |G| tends to infinity.
Original language | English |
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Pages (from-to) | 1755-1807 |
Number of pages | 56 |
Journal | Algebra and Number Theory |
Volume | 15 |
Issue number | 7 |
DOIs | |
Publication status | Published - 1 Nov 2021 |
Bibliographical note
Publisher Copyright:© 2021 Mathematical Sciences Publishers.
Keywords
- Primitive groups
- base sizes
- soluble stabilisers