PT-group

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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