Using transitivity to prove subgroup property satisfaction

A transitive subgroup property is a property $$p$$ such that whenever $$H \le K \le G$$ are groups such that $$K$$ has property $$p$$ in $$G$$ and $$H$$ has property $$p$$ in $$K$$, then $$H$$ has property $$p$$ in $$G$$.

Showing that a subgroup property is a transitive subgroup property is a useful tool in proving various things about it. This article discusses some of the ways transitivity is used.

Also refer the articles proving transitivity and disproving transitivity.

The general idea
Suppose $$H$$ is a subgroup of $$G$$ and $$p$$ is a transitive subgroup property. Then, in order to show that $$H$$ satisfies $$p$$ in $$G$$, it suffices to find an intermediate subgroup $$K$$ such that $$H$$ satisfies $$p$$ in $$K$$ and $$p$$ in $$G$$.

Proving that a subgroup is characteristic
In order to show that a subgroup of a group is characteristic, we often use the fact that characteristic subgroups of characteristic subgroups are characteristic. Thus, for instance:


 * If $$K$$ is a characteristic subgroup of a group $$G$$, applying any subgroup-defining function to $$K$$ also yields a characteristic subgroup. For instance, the center of any characteristic subgroup is characteristic, and so is the commutator subgroup of any characteristic subgroup.
 * Any characteristic subgroup of a subgroup obtained by applying a subgroup-defining function to the whole group is characteristic in the whole group.
 * Applying a subgroup-defining function iteratively or applying one subgroup-defining function after another gives characteristic subgroups.
 * Often, in order to show that a subgroup is characteristic, we show that it is contained in some bigger characteristic subgroup (such as a normal Hall subgroup) and is characteristic in it. Thus, to show that a $$p$$-subgroup of a finite nilpotent group is characteristic in the whole group, it suffices to show that the subgroup is characteristic in the normal $$p$$-Sylow subgroup.