The term quasinormal subgroup was introduced by Øystein Ore in 1937.
are said to commute if HK = KH, that is, any element of the form
Every normal subgroup is quasinormal, because a normal subgroup commutes with every element of the group.
-group for the same (odd) prime has the property that all its subgroups are quasinormal.
If all subgroups are quasinormal, then the group is called an Iwasawa group—sometimes also called a modular group,[1] although this latter term has other meanings.
In any group, every quasinormal subgroup is ascendant.
Every quasinormal subgroup is conjugate permutable.
This follows from the somewhat stronger statement that every conjugate permutable subgroup is subnormal, which in turn follows from the statement that every maximal conjugate permutable subgroup is normal.
In summary, a subgroup H of a finite group G is permutable in G if and only if H is both modular and subnormal in G.[1][2] Permutability is not a transitive relation in general.
The groups in which permutability is transitive are called PT-groups, by analogy with T-groups in which normality is transitive.