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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
View a complete list of group properties
VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

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.
VIEW: Definitions built on this | Facts about this: (facts closely related to PT-group, all facts related to PT-group) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |
View a list of other standard non-basic definitions


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.


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