Permutable subgroup

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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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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 (or quasinormal) if it satisfies the following equivalent conditions:

1. Its product with every subgroup of the group is a subgroup
2. It permutes (or commutes) with every subgroup.
3. It permutes with every cyclic subgroup.

Definition with symbols

A subgroup $H$ of a group $G$ is termed permutable (or quasinormal) in $G$ if it satisfies the following equivalent conditions:

1. For any subgroup $K$ of $G$, $HK$ (the product of subgroups $H$ and $K$) is a group
2. For any subgroup $K$ of $G$, $HK=KH$, i.e., $H$ and $K$ are permuting subgroups.
3. For every $g \in G$, $H$ permutes with the cyclic subgroup generated by $g$. In symbols, for every $h \in H$ and $g \in G$, there exists $h' \in H$ and an integer $n$ such that $hg = g^nh'$.

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)

This subgroup property is a monadic second-order subgroup property, viz., it has a monadic second-order description in the theory of groups
View other monadic second-order subgroup properties

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 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

Conjunction with other properties

In certain kinds of groups:

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 cyclic 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. Groups in which permutability is a transitive relation or, in the finite case, groups in which every subnormal subgroup is permutable are called  PT-groups.

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.
ABOUT THIS PROPERTY: View variations of this property satisfying intermediate subgroup condition | View variations of this property not satisfying intermediate subgroup condition
ABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition

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.

Transfer condition

YES: This subgroup property satisfies the transfer condition: if a subgroup has the property in the whole group, its intersection with any subgroup has the property in that subgroup.
View other subgroup properties satisfying the transfer condition

If $H$ is permutable in $G$, then for any subgroup $K \le G$, $H \cap K$ is permutable in $K$.

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 $H$. For full proof, refer: Permutability satisfies image condition

Upper join-closedness

NO: This subgroup property is not upper join-closed: if a subgroup has the property in intermediate subgroups it need not have the property in their join.

It is possible to have a group $G$, a subgroup $H$ of $G$, and intermediate subgroups $K_1$ and $K_2$ such that $H$ is permutable in both $K_1$ and $K_2$ but $H$ is not permutable in $\langle K_1, K_2 \rangle$. For full proof, refer: Permutability is not upper join-closed

Effect of property operators

The simple group operator

Applying the simple group operator to this property gives: simple group

A group is a simple group if and only if it has no proper nontrivial permutable subgroup. For full proof, refer: Simple iff no proper nontrivial permutable subgroup

Testing

GAP code

One can write code to test this subgroup property in GAP (Groups, Algorithms and Programming), though there is no direct command for it.
View the GAP code for testing this subgroup property at: IsPermutable
View other GAP-codable subgroup properties | View subgroup properties with in-built commands
GAP-codable subgroup property

Although there's no in-built GAP command for testing permutability, a short snippet of code (available at GAP:IsPermutable) can be used to create such a function. This is invoked as follows:

IsPermutable(group,subgroup);

Study of the notion

Mathematical subject classification

Under the Mathematical subject classification, the study of this notion comes under the class: 20E07