Permutable subgroup

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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History

Origin of the concept

The notion of permutable subgroup was introduced when it was observed that there are subgroups that are not normal but still commute with every subgroup.

Origin of the term

This term was introduced by: Ore

Permutable subgroups were initially termed quasinormal subgroups by Oystein Ore in 1937. However, the term permutable subgroup has now gained more currency (since it is more descriptive).

Definition

Symbol-free definition

A subgroup of a group is termed permutable if its product with every subgroup of the group is a subgroup, or equivalently, if it commutes with every subgroup.

Definition with symbols

A subgroup $H$ of a group $G$ is termed permutable if it satisfies the following equivalent conditions:

For any subgroup $K$ of $G$ , $HK$ is a group
For any subgroup $K$ of $G$ , $HK=KH$ .

In terms of the relation implication formalism

The subgroup property of permutability can be expressed in the relation implication formalism as: all $\implies$ permuting subgroups, read as: it permutes with all subgroups.

Relation with other properties

Stronger properties

Normal subgroup

Weaker properties

Nilpotent quotient-by-core subgroup (when we are working with a finite group)
Ascendant subgroup
Subnormal subgroup (when we are working with a finite group)
Automorph-permutable subgroup
Conjugate-permutable subgroup
Sylow-permutable subgroup

Relation with normality

Every normal subgroup is permutable, but the converse is not true. In fact, there are groups in which every subgroup is permutable, but where every subgroup is not normal. These are called quasi-Hamiltonian groups. In fact, any extension of a cyclic group of prime power order by another cycilc group of prime power order is quasi-Hamiltonian.

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
ABOUT THIS PROPERTY: View variations of this property that are transitive|View variations of this property that are not transitive
ABOUT TRANSITIVITY: View a complete list of subgroup properties that are not transitive|View facts related to transitivity of subgroup properties | View a survey article on disproving transitivity

In fact, if it were, then every subnormal subgroup would be permutable, which is clearly not the case.

Trimness

This subgroup property is trim -- it is both trivially true (true for the trivial subgroup) and identity-true (true for a group as a subgroup of itself).
View other trim subgroup properties | View other trivially true subgroup properties | View other identity-true subgroup properties

Both the whole group, and the trivial subgroup, are permutable.

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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Permutability satisfies the intermediate subgroup condition. In other words, if $H$ is a permutable subgroup of $G$ , $H$ is also a permutable subgroup of any subgroup $K$ between $H$ and $G$ .

Intersection-closedness

This subgroup property is not intersection-closed, viz., it is not true that an intersection of subgroups with this property must have this property.
Read an article on methods to prove that a subgroup property is not intersection-closed

An example given by Ito shows that an intersection of permutable subgroups need not be intersection-closed. Further information: Permutability is not intersection-closed

Join-closedness

YES: This subgroup property is join-closed: an arbitrary (nonempty) join of subgroups with this property, also has this property.
ABOUT THIS PROPERTY: View variations of this property that are join-closed | View variations of this property that are not join-closed
ABOUT JOIN-CLOSEDNESS: View all join-closed subgroup properties (or, strongly join-closed properties) | View all subgroup properties that are not join-closed | Read a survey article on proving join-closedness | Read a survey article on disproving join-closedness

The subgroup generated by a family of permutable subgroups is permutable.

Inverse image condition

This subgroup property satisfies the inverse image condition. In other words, the inverse image under any homomorphism of a subgroup satisfying the property also satisfies the property. In particular, this property satisfies the transfer condition and intermediate subgroup condition.

If $f:G\to H$ is a homomorphism and $P$ is a permutable subgroup of $H$ , then $f^{-1}(P)$ is a permutable subgroup of $G$ .

Image condition

YES: This subgroup property satisfies the image condition, i.e., under any surjective homomorphism, the image of a subgroup satisfying the property also satisfies the property
View other subgroup properties satisfying image condition

If $f:G\to H$ is a surjective homomorphism and $K$ is a permutable subgroup of $G$ , then $f(K)$ is a permutable subgroup of $H$ .

Definition links

References

Old, recent and new results on quasinormal subgroups