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 such that whenever
are groups such that
has property
in
and
has property
in
, then
has property
in
.
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 is a subgroup of
and
is a transitive subgroup property. Then, in order to show that
satisfies
in
, it suffices to find an intermediate subgroup
such that
satisfies
in
and
in
.
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
is a characteristic subgroup of a group
, applying any subgroup-defining function to
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
-subgroup of a finite nilpotent group is characteristic in the whole group, it suffices to show that the subgroup is characteristic in the normal
-Sylow subgroup.