PT-group

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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This article is about a definition in group theory that is standard among the group theory community (or sub-community that dabbles in such things) but is not very basic or common for people outside.
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Definition

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

In terms of the subgroup property collapse operator

This group property can be defined in terms of the collapse of two subgroup properties. In other words, a group satisfies this group property if and only if every subgroup of it satisfying the first property (permutable subgroup) satisfies the second property (right-transitively permutable subgroup), and vice versa.
View other group properties obtained in this way

It can be expressed using the following collapses:

Relation with other properties

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