Permutable subgroup

How permutable subgroups came about

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).

Definitions

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 for every subgroup $K$, $HK=KH$.

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.

Property theory of permutability

Transitivity

Permutability is identity-true but not transitive. There is no nontrivial information about the left transiter and the right transiter of permutability.

Identity-true

Permutability is an identity-true subgroup property, in fact, it is a trim subgroup property.

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-closed

It is not clear whether the intersection of permutable subgroups is permutable.

Subgroup-generation-closed

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

• Old, recent and new results on quasinormal subgroups