Finite solvable not implies p-normal
This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., finite solvable group) need not satisfy the second subgroup property (i.e., p-normal group)
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Example of the symmetric group of degree four
Further information: symmetric group:S4
Let be the symmetric group on the set and let . Then:
- is solvable.
- is not -normal: The center of a -Sylow subgroup is not weakly closed in it. For instance, consider the -Sylow subgroup:
The center of this is:
This is not weakly closed in . For instance, it is conjugate in to the subgroup generated by , which is another subgroup inside .