### Some interesting statistics concerning finite primitive permutation groups

Combinatorics Seminar

17th December 2019, 11:00 am – 12:00 pm

Fry Building, 2.04

Let G be a finite permutation group on a set X. A BASE for G is a subset Y of X such that G_(Y), the pointwise-stabilizer of Y in G, is trivial. There has been a long history of studying how small a base can be for different classes of group G. We will discuss some variants of this study, particularly focusing on upper bounds for primitive groups: in particular, we want to know how big a minimal base can be, how big an irredundant base can be, and how big an independent set can be. (The precise definition of these three notions will be given in the seminar.)

Our interest in these statistics stems from their connection to another statistic -- the RELATIONAL COMPLEXITY of a finite permutation group. This last statistic was introduced in the 1990's by Greg Cherlin in work applying certain model theoretic ideas of Lachlan. In particular the relational complexity of a permutation group gives an idea of the "efficiency" with which the group can be represented as the automorphism group of a homogeneous relational structure.

This is joint work with Bianca Loda and Pablo Spiga.

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