Pronormality does not satisfy transfer condition
This article gives the statement, and possibly proof, of a subgroup property (i.e., pronormal subgroup) not satisfying a subgroup metaproperty (i.e., transfer condition).
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Related metaproperties satisfied by pronormality
- Pronormality satisfies image condition
- Pronormality satisfies intermediate subgroup condition
- Pronormality is quotient-transitive
Related metaproperties not satisfied by pronormality
Other facts relating pronormality and the transfer condition
- Transfer-closed pronormal subgroup: A subgroup whose intersection with any other subgroup is pronormal in that other subgroup.
Example of the symmetric group
Further information: symmetric group:S4
Let be the symmetric group on the set . Consider subgroups of as follows:
Then, we have:
- is not pronormal in : Indeed, the subgroup is conjugate to it in , but not in the subgroup they generate. Another way of seeing this is that is 2-subnormal but not normal in , hence it cannot be pronormal in .
- is pronormal in : is a subgroup whose join with any distinct conjugate is the whole group. Thus, by fact (1), is pronormal in .