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]
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]
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 of a group is termed conjugate-permutable if for all in , or equivalently, if is a group for all in .
(here is a conjugate subgroup to in ).
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 is conjugate-permutable in a group if the following first-order sentence is satisfied:
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 permuting subgroups. In other words is conjugate-permutable in iff any conjugate subgroup to , permutes with .
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 such that is conjugate-permutable in and is conjugate-permutable in , but is not conjugate-permutable in . |
intermediate subgroup condition | Yes | conjugate-permutability satisfies intermediate subgroup condition | If is a conjugate-permutable subgroup of , and is any intermediate subgroup of , then is conjugate-permutable in . |
conjugate-join-closed subgroup property | Yes | conjugate-permutability is conjugate-join-closed | If is a conjugate-permutable subgroup of , then a join of conjugate subgroups to is again conjugate-permutable. |
finite-join-closed subgroup property | No | conjugate-permutability is not finite-join-closed | It is possible to have a group and conjugate-permutable subgroup of such that the join of subgroups 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 and conjugate-permutable subgroup of such that the intersection of subgroups is not a conjugate-permutable subgroup. |
directed union-closed subgroup property | Yes | conjugate-permutability is directed union-closed | Suppose is a directed set of subgroups of a group , and every is conjugate-permutable in . Then, the directed union of the s is also conjugate-permutable in . |
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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