PT-group

Symbol-free definition
A group is termed a PT-group if every subpermutable subgroup of the group is a permutable subgroup. In other words, a group is termed a PT-group if every permutable subgroup of a permutable subgroup is permutable.

For finite groups, this is the same as requiring that every subnormal subgroup be permutable.

Definition with symbols
A group $$G$$ is termed a PT-group if, whenever $$H$$ is a permutable subgroup of $$G$$ and $$K$$ is a permutable subgroup of $$H$$, then $$K$$ is a permutable subgroup of $$G$$.

Formalisms
It can be expressed using the following collapses:


 * defining ingredient::subpermutable subgroup = permutable subgroup
 * permutable subgroup = right-transitively permutable subgroup

Stronger properties

 * T-group (in the case of finite groups): In the finite case, since every subpermutable subgroup is subnormal, every T-group is a PT-group