Conjugate-permutable subgroup

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).

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 a subgroup.

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is termed conjugate-permutable if $$HH^g = H^gH$$ for all $$g$$ in $$G$$, or equivalently, if $$HH^g$$ is a group for all $$g$$ in $$G$$.

(here $$H^g = g^{-1}Hg$$ is a conjugate subgroup to $$H$$ in $$G$$).

Formalisms
A subgroup $$H$$ is conjugate-permutable in a group $$G$$ if the following first-order sentence is satisfied:

$$\forall g \in G, \forall x,y \in H \exists a,b \in H. (xgyg^{-1} = gag^{-1}b)$$

The property of being conjugate-permutable can be encoded as conjugate subgroups $$\implies$$ permuting subgroups. In other words $$H$$ is conjugate-permutable in $$G$$ iff any conjugate subgroup to $$H$$, permutes with $$H$$.

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

Stronger properties

 * Weaker than::Normal subgroup
 * Weaker than::Permutable subgroup
 * Weaker than::2-subnormal subgroup:
 * Weaker than::Automorph-permutable subgroup

Weaker properties

 * Subnormal subgroup (for finite groups):
 * Stronger than::Descendant subgroup:

Opposite properties

 * Self-conjugate-permutable subgroup: A subgroup is both conjugate-permutable and self-conjugate-permutable iff it is normal.

Metaproperties
The property of being conjugate-permutable is nottransitive. For finite groups, its subordination is in fact the same as the property of being subnormal.

If $$H$$ is a conjugate-permutable subgroup of $$G$$, and $$K$$ is any intermediate subgroup of $$G$$, then $$H$$ is conjugate-permutable in $$K$$.

If $$H$$ is a conjugate-permutable subgroup of $$G$$, then a join of conjugate subgroups to $$H$$ is again conjugate-permutable.

A join of conjugate-permutable subgroups of a group is not necessarily conjugate-permutable.

An intersection of conjugate-permutable subgroups of a group is not necessarily conjugate-permutable.

The union of a directed set of conjugate-permutable subgroups is again conjugate-permutable.

Transiters and residuals

 * The right transiter of conjugate-permutability is the balanced subgroup property corresponding to subgroup-conjugating automorphisms, namely the property of being a SCAB-subgroup.
 * 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.

Article links

 * Weblink for Conjugate-permutable subgroups by Tuval Foguel
 * Weblink for Groups with all cyclic subgroups conjugate-permutable by Tuval Foguel