# Analogue of Thompson transitivity theorem fails for abelian subgroups of rank two

## Statement

It is possible to choose a finite group $G$ and a prime number $p$ such that $G$ is a group in which every p-local subgroup is p-constrained, such that:

There exists a subgroup $A$ of $G$ that is maximal among abelian normal subgroups in some $p$-Sylow subgroup of $G$, such that $A$ has rank two, and there is a prime $q \ne p$ such that $C_G(A)$ is not transitive on the collection of maximal $A$-invariant $q$-subgroups of $G$.

In other words, the analogue of the Thompson transitivity theorem fails if we drop the assumption of rank at least three.