On generalizing transitivity, persistence, and sensitivity

Abstract

A subgroup property $\alpha$ is transitive in a group $G$ if $U \alpha V$ and $V \alpha G$ imply that $U \alpha G$ whenever $U \le V \le G$, and $\alpha$ is persistent in $G$ if $U \alpha G$ implies that $U \alpha V$ whenever $U \le V \le G$. Even though a subgroup property $\alpha$ may be neither transitive nor persistent, a given subgroup $U$ may have the property that each $\alpha$-subgroup of $U$ is an $\alpha$-subgroup of $G$, or that each $\alpha$-subgroup of $G$ in $U$ is an $\alpha$-subgroup of $U$. We call these subgroup properties $\alpha$-transitivity and $\alpha$-persistence, respectively. We introduce and develop the notions of $\alpha$-transitivity and $\alpha$-persistence, and we establish how the former property is related to $\alpha$-sensitivity. In order to demonstrate how these concepts can be used, we apply the results to the cases in which $\alpha$ is replaced with “normal” and the “cover-avoidance property.” We also suggest ways in which the theory can be developed further.