Conjugacy-closed and Hall not implies retract
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., conjugacy-closed Hall subgroup) need not satisfy the second subgroup property (i.e., Hall retract)
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A conjugacy-closed Hall subgroup (i.e., a Hall subgroup that is conjugacy-closed: any two elements in the subgroup that are conjugate in the whole group are conjugate in the subgroup) need not be a Retract (?). Specifically, there need not be a Normal Hall subgroup (?) that is a complement to it.
- Conjugacy-closed and Sylow implies retract
- Conjugacy-closed Abelian Sylow implies retract
- Conjugacy-closed Abelian Hall implies retract
Example of Sylow complements in symmetric groups
Further information: Symmetric group on subset is conjugacy-closed
Let be a prime equal to or more. Then, let and . Define and . Then:
- conjugacy-closed: The group is a conjugacy-closed subgroup of the group by the above fact.
- Hall: is also a Hall subgroup (in fact, it is a Hall -subgroup) of .
- Not a retract: For this, observe that for , the only normal subgroups of are the whole group, the trivial subgroup, and the alternating group. None of these have order , and thus, cannot be realized as a retract.