Pronormality is not centralizer-closed
This article gives the statement, and possibly proof, of a subgroup property (i.e., pronormal subgroup) not satisfying a subgroup metaproperty (i.e., centralizer-closed subgroup property).
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- Pronormality is normalizer-closed: In fact, normalizer of pronormal implies abnormal
- Pronormality is not commutator-closed
- Automorph-conjugacy is centralizer-closed
- Isomorph-conjugacy is not centralizer-closed
Example of the symmetric group
Suppose is the symmetric group on the set and is a -Sylow subgroup of , given by:
Then, is a pronormal subgroup of , because it is a Sylow subgroup and Sylow implies pronormal. On the other hand, we have:
This is not pronormal, because it is conjugate to the subgroup via the permutation , but these are not conjugate in the subgroup they generate.
Let be the symmetric group on the set .