### Finitary Permutation Groups Study Notes

During my PhD, I participated in various study groups, one of which was the Combinatorics Study Group. These students, postdoc researchers and academics would spend Wednesday afternoons counting things, studying graphs and other objects, and studying groups. They would then adjoin to the Old Globe on Mile End Road (sadly now a Ladbrokes) where the real ales would be drained. One on occasion, we even drank the Eagle Smooth before going to another pub because we finished that as well.

Permutations arise in the study of Combinatorics and Finitary Groups were key to my PhD study. I put these notes together as a knowledge session for the group. There is a proof that uses the Axiom of Choice and if I recall correctly, this led to a pub discussion where one of the senior academics declared they didn’t believe it. (Not all of their infinite vector spaces have bases either…)

I cannot actually remember when I wrote these, but they site Peter‘s Permutation Groups book so it must be after 1999.

#### Abstract

A **finitary permutation group** is a natural generalization of a finite permutation group. The structure of a transitive finitary permutation group is surprisingly simple when its degree is infinite. Here we study primitivity, following P. M. Neumannâ€™s work in the 1970s. We also study generalized solubility conditions on these groups. These notes arose from lectures aimed at an audience who had seen some basic permutation group theory, but little abstract group theory.

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