Transitively normal not implies conjugacy-closed normal

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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., transitively normal subgroup) need not satisfy the second subgroup property (i.e., conjugacy-closed normal subgroup)
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Statement

A transitively normal subgroup of a group (a subgroup with the property that every normal subgroup of it is normal in the whole group) need not be conjugacy-closed in the whole group. In other words, there may be elements in the subgroup that are conjugate in the whole group but not conjugate in the subgroup.

Definitions used

Transitively normal subgroup

Further information: Transitively normal subgroup

A subgroup H of a group G is termed transitively normal in G if whenever K is a normal subgroup of H, K is normal in G.

Alternatively, the property of being transitively normal can be described using this function restriction expression:

Inner automorphism \to Normal automorphism

In other words, H is transitively normal in G if and only if every inner automorphism of G restricts to a normal automorphism of H (an automorphism that preserves normal subgroups).

Conjugacy-closed normal subgroup

Further information: Conjugacy-closed subgroup, conjugacy-closed normal subgroup

A subgroup H of a group G is termed conjugacy-closed in G if whenever two elements of H are conjugate in G, they are conjugate in H. A conjugacy-closed normal subgroup is a subgroup that is both conjugacy-closed and normal.

Alternatively, the property of being conjugacy-closed normal can be described using this function restriction expression:

Inner automorphism \to Class-preserving automorphism

In other words, H is conjugacy-closed normal in G if and only if every inner automorphism of G restricts to a class-preserving automorphism of H: an automorphism that preserves conjugacy classes of elements.

Related facts

An equivalent fact

Normal not implies class-preserving: A normal automorphism of a group (i.e., an automorphism that preserves normal subgroups) need not be class-preserving (i.e., it may not send every element to a conjugate element).

Facts used

  1. Normal not implies class-preserving

Proof

Proof using fact (1)

We show that if H is a group with a normal automorphism \sigma that is not class-preserving, then we can construct a group G containing H as a transitively normal subgroup that is not conjugacy-closed normal.

Let K be the cyclic subgroup of \operatorname{Aut}(G) generated by \sigma, and define G = H \rtimes K with the specified action. Then:

  • H is transitively normal in G: Any inner automorphism in G is generated by \sigma and inner automorphisms in H. Each of these preserve normal subgroups of H, so H is transitively normal in G.
  • H is not conjugacy-closed normal in G: Conjugation by \sigma \in G restricts to the automorphism \sigma of H, that is not a class-preserving automorphism by assumption.

A concrete example

The smallest example is where the group is the symmetric group on three letters, and the subgroup is the alternating group on three letters. The subgroup is a simple normal subgroup, so it is in particular transitively normal. On the other hand, it is not conjugacy-closed: the two elements of order three are conjugate by a transposition in the symmetric group but are not conjugate within the alternating group.

Note that this corresponds to fact (1) if we start with the inverse map as a normal automorphism of the cyclic group of order three that is not class-preserving.