Pronormality is not commutator-closed
This article gives the statement, and possibly proof, of a subgroup property (i.e., pronormal subgroup) not satisfying a subgroup metaproperty (i.e., commutator-closed subgroup property).
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- Normality is commutator-closed
- Characteristicity is commutator-closed
- Endo-invariance implies commutator-closed
- Join of subnormal subgroups is subnormal iff their commutator is subnormal
Example of the symmetric group of degree four
Further information: symmetric group:S4
Let be the symmetric group on the set . Let be a -Sylow subgroup of , say:
Then, is isomorphic to a dihedral group of order eight. .
- is pronormal in : In fact, is a Sylow subgroup of , and Sylow implies pronormal.
- is not pronormal in : This subgroup is conjugate to by the permutation , but these two subgroups are not conjugate in the subgroup they generate.