Automorph-conjugacy is not finite-join-closed
This article gives the statement, and possibly proof, of a subgroup property (i.e., automorph-conjugate 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 automorph-conjugate subgroups need not be automorph-conjugate.
For the Hall subgroups example
- Sylow implies automorph-conjugate
- Hall implies join of Sylow subgroups
- Hall not implies automorph-conjugate
The Hall subgroups example
The proof follows directly from facts (1)-(3).