Using transitivity to prove subgroup property satisfaction
This is a survey article related to:subgroup property satisfaction
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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.
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.