Conjugacy-closed normal not implies central factor
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 normal subgroup) need not satisfy the second subgroup property (i.e., central factor)
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Contents
Statement
Property-theoretic statement
The subgroup property of being a conjugacy-closed normal subgroup is not stronger than the subgroup property of being a central factor.
Definitions used
Conjugacy-closed normal subgroup
Further information: Conjugacy-closed normal subgroup
- Hands-on definition: A subgroup of a group is termed a conjugacy-closed normal subgroup if is normal in , and, whenever two elements of are conjugate in , they are conjugate in .
- Definition in terms of function restriction expressions: The property of being conjugacy-closed normal can be expressed using the following function restriction expression:
Inner automorphism Class-preserving automorphism
In other words, is conjugacy-closed normal in if every inner automorphism of restricts to a class-preserving automorphism of -- an automorphism that preserves conjugacy classes.
Central factor
Further information: Central factor
- Hands-on definition: A subgroup of a group is termed a central factor if .
- Definition in terms of function restriction expressions: The property of being a central factor can be expressed as:
Inner automorphism Inner automorphism
In other words, is a central factor in if every inner automorphism of restricts to an inner automorphism of .
Related facts
An equivalent fact
- Class-preserving not implies inner: Not every class-preserving automorphism of a group is inner.
Facts used
Proof
Proof using fact (1)
We shall prove the following. If possesses a class-preserving automorphism that is not inner, we can find a group containing as a conjugacy-closed normal subgroup that is not a central factor.
Let be the cyclic subgroup of generated by , and let , with the specified action of on .
- is conjugacy-closed normal in : Indeed, any inner automorphism of is generated by and inner automorphisms from , and each of these restrict to class-preserving automorphisms of .
- is not a central factor of : Indeed, conjugation by the element induces the automorphism on , that is not an inner automorphism.
An infinite group example
Further information: Finitary symmetric group is conjugacy-closed in symmetric group, Finitary symmetric group is centralizer-free in symmetric group
In the symmetric group on an infinite set , the subgroup comprising the finitary permutations (those permutations that move only finitely many elements) is a conjugacy-closed subgroup: two finitary permutations that are conjugate by some permutation are conjugate by a finitary permutation. Further, it is a normal subgroup: any conjugate to a finitary permutation is again finitary.
Thus, the finitary symmetric group is a conjugacy-closed normal subgroup in the whole symmetric group . On the other hand, it is not a central factor. To see this, observe that by the first definition of central factor, it suffices to show that the centralizer in the whole symmetric group, of the finitary symmetric group, is trivial.
Let's do this. Suppose commutes with every element in . Now, pick . We will show that .
First, since has at least three elements, we can pick such that . Let be a transposition.
Since commutes with , we are forced to conclude that either or . Since by assumption, , completing the proof.