Conjugate-permutable subgroup

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History

Origin

Both the term and the concept of conjugate-permutable subgroups arose in the paper Conjugate-permutable subgroups by Tuval Foguel. Foguel introduced this notion by observing that the proof that permutable subgroups are subnormal actually generalizes to showing that conjugate-permutable subgroups are subnormal (in finite groups).

Definition

Symbol-free definition

A subgroup of a group is termed conjugate-permutable if it permutes with every conjugate of itself, or equivalently, if its product with every conjugate of it is the whole group.

Definition with symbols

A subgroup $H$ of a group $G$ is termed conjugate-permutable if $H{H}^g=H^{g}H$ for all $g$ in $G$, or equivalently, if $H{H}^g$ is a group for all $g$ in $G$.

In terms of the permutability operator

The property of being conjugate-permutable is obtained by applying the permutability operator to the subgroup pair property of being conjugate.

Relation with other properties

Stronger properties

• Normal subgroup
• Permutable subgroup
• Automorph-permutable subgroup

Weaker properties

• Subnormal subgroup (for finite groups) -- For full proof, refer: Conjugate-permutable implies subnormal (finite groups)

Metaproperties

Transitivity

NO: This subgroup property is not transitive: a subgroup with this property in a subgroup with this property, need not have the property in the whole group
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The property of being conjugate-permutable is nottransitive. For finite groups, its subordination is in fact the same as the property of being subnormal.

Intermediate subgroup condition

YES: This subgroup property satisfies the intermediate subgroup condition: if a subgroup has the property in the whole group, it has the property in every intermediate subgroup.
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If $H$ is a conjugate-permutable subgroup of $G$, and $K$ is any intermediate subgroup of $G$, then $H$ is conjugate-permutable in $K$.

Property operators

Maximal operator

Any maximal conjugate-permutable subgroup is either normal or permutably contranormal. Here, a subgroup is termed permutably contranormal if its product with any conjugate of it is the whole group.

We can prove that there are no proper permutably contranormal subgroups of finite index. Thus, any maximal conjugate-permutable subgroup of finite index in a group is normal.

In particular, the submax operator applied to the property of being conjugate-permutable with finite index, gives the property of subnormality.

In particular, for finite groups, every conjugate-permutable subgroup is subnormal.

Transiters and residuals

• The right transiter of conjugate-permutability is the balanced subgroup property corresponding to subgroup-conjugating automorphisms.
• The left residual by normality is the property of being automorph-permutable. Incidentally, this also shows that any 2-subnormal subgroup of a group is conjugate-permutable.