Transitivizer

Symbol-free definition
A subgroup property is said to be a transitivizer of another subgroup property if it is transitive and its conjunction with the other subgroup property is also transitive.

Definition with symbols
A subgroup property $$p$$ is termed a transitivizer for a subgroup property $$q$$ if $$p$$ is transitive and the logical conjunction of $$p$$ with $$q$$ is also transitive.