Pronormality is not finite-join-closed
This article gives the statement, and possibly proof, of a subgroup property (i.e., pronormal subgroup) not satisfying a subgroup metaproperty (i.e., finite-join-closed subgroup property).This also implies that it does not satisfy the subgroup metaproperty/metaproperties: Join-closed subgroup property (?), .
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A join of finitely many pronormal subgroups of a group need not be pronormal.
- Procharacteristicity is not finite-join-closed
- Left residual of pronormal by normal is procharacteristic
- Finite-join-closedness is left residual-preserved
The proof follows directly by combining facts (1)-(3).