Permutable subgroup

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
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).

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
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.

Stronger properties

 * Weaker than::Normal subgroup:

Weaker properties

 * Stronger than::Ascendant subgroup
 * Stronger than::Automorph-permutable subgroup
 * Stronger than::Conjugate-permutable subgroup
 * Stronger than::Sylow-permutable subgroup
 * Stronger than::Subnormal-permutable subgroup
 * Stronger than::Modular subgroup:
 * Stronger than::Elliptic subgroup

Conjunction with other properties
In certain kinds of groups:


 * Permutable subgroup of finite group is a permutable subgroup inside a finite group.

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
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.

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

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.

An example given by Ito shows that an intersection of permutable subgroups need not be intersection-closed.

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

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

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 $$G$$

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 $$H$$.

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 $$\langle K_1, K_2 \rangle$$.

Effect of property operators
A group is a simple group if and only if it has no proper nontrivial permutable subgroup.

Testing
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);

Textbook references

 * , Pages 14-16, Permutable subgroups and normal subgroups
 * , Page 43, Modular and permutable subgroups

Articles

 * Old, recent and new results on quasinormal subgroups