Automorph-conjugacy is not finite-conjugate-intersection-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-conjugate-intersection-closed subgroup property).
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Example in the symmetric group
Further information: symmetric group:S6
Let be the symmetric group on six letters: . Let be the following 2-Sylow subgroups of :
In other words, is the internal direct product of a 2-Sylow subgroup on with the 2-Sylow subgroup on , while is the internal direct product of the 2-Sylow subgroup on with a 2-Sylow subgroup on .
The intersection is given by:
Now, note that:
- Both and are automorph-conjugate, because they are both Sylow subgroups, and Sylow implies automorph-conjugate. Moreover, they are conjugate to each other, since Sylow implies order-conjugate.
- is not automorph-conjugate. To see this, note that has an outer automorphism that sends transpositions to triple transpositions. Under this automorphism, goes to a subgroup of that contains three commuting triple transpositions. If this is conjugate to , then should also contain three commuting triple transpositions. But it doesn't.