Almost group theory

A note on FC-nilpotency

“…The result presented here generalizes some previous cases due to Wagner [9, Proposition 4.4.10] for groups in simple theories, as well as in [5] for groups satisfying a uniform chain condition on centralizers up to bounded index. Our proof involves some machinery on FC-centralizers recently obtained by the first author in [5] using techniques from model theory. Finally, concerning the FC-solvable case note that the situation is more straightforward and an easy argument is given at the end of the paper.…”

Pseudofinite primitive permutation groups acting on one-dimensional sets

“…Then y ∈ L x if and only if dim(x G (x 1 ) g ) = 0 if and only if (x g −1 ) Gx 1 = 0 if and only if there is g ∈ G such that (x 1 ) g = y and Proof. If G has a non-trivial normal abelian subgroup, then G has a definable finiteby-abelian subgroup A, which is normal in G and contains H, by [3,Theorem 3.3(1)]. Since A ′ is definable and of dimension 0, by definable primitivity, A ′ is trivial, hence A is abelian.…”

“…On the other hand, the chain condition on centralizers focuses more on the combinatoric properties that a tame theory should have. This condition itself decreases the complexity of groups and gives some nice structural theorems for definable subgroups (see [3] for more details). However, classical tame model theory usually has more powerful well-developped tools for analysis, for example the Indecomposability Theorem in supersimple theories.…”

“…[3, Theorem 2.10] Let H and K be two definable subgroups of G. Then H C G (K) if and only if K C G (H).…”

“…If H is abelian. Then by [3,Theorem 3.3(1)] G has a definable finite-by-abelian normal subgroup A ≥ H. By definably primitivity, A is abelian. By Lemma 3.2, A acts regularly on X.…”