# Using transitivity to prove subgroup property satisfaction

This is a survey article related to:subgroup property satisfaction
View other survey articles about 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

Further information: Characteristicity is transitive

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.