Conjugacy-closed and Hall not implies retract

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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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Statement

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.

Related facts

Proof

Example of Sylow complements in symmetric groups

Further information: Symmetric group on subset is conjugacy-closed

If S \subseteq T are sets, then the symmetric group on S embeds as a conjugacy-closed subgroup of the symmetric group on T. In other words, if two elements of \operatorname{Sym}(S) are conjugate in \operatorname{Sym}(T), they are also conjugate in \operatorname{Sym}(S).

Let p be a prime equal to 5 or more. Then, let S = \{ 1,2,3,\dots,p-1 \} and T = \{ 1,2,3,\dots,p \}. Define S_{p-1} = \operatorname{Sym}(S) and S_p = \operatorname{Sym}(T). Then:

  • conjugacy-closed: The group S_{p-1} = \operatorname{Sym}(S) is a conjugacy-closed subgroup of the group S_p = \operatorname{Sym}(T) by the above fact.
  • Hall: S_{p-1} is also a Hall subgroup (in fact, it is a Hall p'-subgroup) of S_p.
  • Not a retract: For this, observe that for p \ge 5, the only normal subgroups of S_p are the whole group, the trivial subgroup, and the alternating group. None of these have order p, and thus, S_{p-1} cannot be realized as a retract.