Pronormality is not finite-join-closed

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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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Statement

A join of finitely many pronormal subgroups of a group need not be pronormal.

Facts used

  1. Procharacteristicity is not finite-join-closed
  2. Left residual of pronormal by normal is procharacteristic
  3. Finite-join-closedness is left residual-preserved

Proof

Property-theoretic proof

The proof follows directly by combining facts (1)-(3).

Hands-on proof: example of the skew-linear group

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