Conjugate-permutable subgroup

This article defines a term that has been used or referenced in a journal article or standard publication, but may not be generally accepted by the mathematical community as a standard term.[SHOW MORE]

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This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties [SHOW MORE]
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This is a variation of permutability|Find other variations of permutability |

History

Origin

This term was introduced by: Tuval Foguel

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 a subgroup.

Definition with symbols

A subgroup $H$ of a group $G$ is termed conjugate-permutable if $HH^{g}=H^{g}H$ 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

BEWARE! This section of the article uses terminology local to the wiki, possibly without giving a full explanation of the terminology used (though efforts have been made to clarify terminology as much as possible within the particular context)

First-order description

This subgroup property is a first-order subgroup property, viz., it has a first-order description in the theory of groups.
View a complete list of first-order subgroup properties

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

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

Relation implication expression

This subgroup property is a relation implication-expressible subgroup property: it can be defined and viewed using a relation implication expression
View other relation implication-expressible subgroup properties

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 $H$ .

In terms of the permutability operator

This property is obtained by applying the permutability operator to the property: conjugate subgroups
View other properties obtained by applying 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
2-subnormal subgroup: For full proof, refer: 2-subnormal implies conjugate-permutable
Automorph-permutable subgroup

Weaker properties

Subnormal subgroup (for finite groups): For proof of the implication, refer Conjugate-permutable implies subnormal (finite groups) and for proof of its strictness (i.e. the reverse implication being false) refer Subnormal not implies conjugate-permutable.
Descendant subgroup: For full proof, refer: Conjugate-permutable implies descendant

Opposite properties

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

Metaproperties

Metaproperty name	Satisfied?	Proof	Statement with symbols
transitive subgroup property	No	conjugate-permutability is not transitive	It is possible to have groups $H\leq K\leq G$ such that $H$ is conjugate-permutable in $K$ and $K$ is conjugate-permutable in $G$ , but $H$ is not conjugate-permutable in $G$ .
intermediate subgroup condition	Yes	conjugate-permutability satisfies intermediate subgroup condition	If $H$ is a conjugate-permutable subgroup of $G$ , and $K$ is any intermediate subgroup of $G$ , then $H$ is conjugate-permutable in $K$ .
conjugate-join-closed subgroup property	Yes	conjugate-permutability is conjugate-join-closed	If $H$ is a conjugate-permutable subgroup of $G$ , then a join of conjugate subgroups to $H$ is again conjugate-permutable.
finite-join-closed subgroup property	No	conjugate-permutability is not finite-join-closed	It is possible to have a group $G$ and conjugate-permutable subgroup $H,K$ of $G$ such that the join of subgroups $\langle H,K\rangle$ is not a conjugate-permutable subgroup.
finite-intersection-closed subgroup property	No	conjugate-permutability is not finite-intersection-closed	It is possible to have a group $G$ and conjugate-permutable subgroup $H,K$ of $G$ such that the intersection of subgroups $H\cap K$ is not a conjugate-permutable subgroup.
directed union-closed subgroup property	Yes	conjugate-permutability is directed union-closed	Suppose $(H_{i})_{i\in I}$ is a directed set of subgroups of a group $G$ , and every $H_{i}$ is conjugate-permutable in $G$ . Then, the directed union of the $H_{i}$ s is also conjugate-permutable in $G$ .

Effect of property operators

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. For full proof, refer: Right-transiter of conjugate-permutable is SCAB
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. For full proof, refer: left residual of conjugate-permutable by normal is automorph-permutable

References

Conjugate-permutable subgroups by Tuval Foguel, Journal of Algebra, ISSN 00218693, Volume 191, Article number JA966924, Page 235 - 239(Year 1997): This paper defines the notion of conjugate-permutable subgroup, and proves (among other things) that conjugate-permutable subgroups in finite groups are subnormal.^{Weblink for homepage copy}^{More info}
Groups with all cyclic subgroups conjugate-permutable groups by Tuval Foguel, Journal of Group Theory, ISSN 14435883 (print), ISSN 14434446 (online), Volume 2, Page 47 - 51(Year 1999): ^{Weblink for homepage copy}^{More info}
Groups with conjugate-permutable conditions by Shirong Li and Zhongchuan Meng, Mathematical Proceedings of the Royal Irish Academy

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