A note on pseudofinite groups of finite centraliser dimension

We give a structural theorem for pseudofinite groups of finite centraliser dimension. As a corollary, we observe that there is no finitely generated pseudofinite group of finite centraliser dimension.Peer reviewe

Profinite groups with NIP theory and p-adic analytic groups

We consider profinite groups as 2‐sorted first‐order structures, with a group sort, and a second sort that acts as an index set for a uniformly definable basis of neighbourhoods of the identity. It is shown that if the basis consists of all open subgroups, then the first‐order theory of such a structure is NIP (that is, does not have the independence property) precisely if the group has a normal subgroup of finite index that is a direct product of finitely many compact p ‐adic analytic groups, for distinct primes p . In fact, the condition NIP can here be weakened to NTP 2 . We also show that any NIP profinite group, presented as a 2‐sorted structure, has an open prosoluble normal subgroup

Model theory of finite and pseudofinite groups

This is a survey, intended both for group theorists and model theorists, concerning the structure of pseudofinite groups, that is, infinite models of the first-order theory of finite groups. The focus is on concepts from stability theory and generalisations in the context of pseudofinite groups, and on the information this might provide for finite group theory