Conjugate-permutable subgroup

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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 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

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

Weaker properties

Opposite properties

Metaproperties

Metaproperty name Satisfied? Proof Statement with symbols
transitive subgroup property No conjugate-permutability is not transitive It is possible to have groups H \le K \le 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_is is also conjugate-permutable in G.

Effect of property operators

Transiters and residuals

References

External links

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Definition links

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