Left-transitively permutable subgroup
From Groupprops
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This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]
Definition
Symbol-free definition
A subgroup of a group is termed left-transitively permutable if whenever the group is embedded as a permutable subgroup of some bigger group, the subgroup is also permutable inside the bigger group.
Definition with symbols
A subgroup of a group
is termed left-transitively permutable if for any embedding of
in some group
, such that
is permutable inside
,
is also permutable inside
.
Relation with other properties
Weaker properties
- Characteristic subgroup: For full proof, refer: Left-transitively permutable implies characteristic
- Normal subgroup
- Permutable subgroup
Metaproperties
Transitivity
This subgroup property is transitive: a subgroup with this property in a subgroup with this property, also has this property in the whole group.
ABOUT THIS PROPERTY: View variations of this property that are transitive | View variations of this property that are not transitive
ABOUT TRANSITIVITY: View a complete list of transitive subgroup properties|View a complete list of facts related to transitivity of subgroup properties |Read a survey article on proving transitivity
Trimness
This subgroup property is trim -- it is both trivially true (true for the trivial subgroup) and identity-true (true for a group as a subgroup of itself).
View other trim subgroup properties | View other trivially true subgroup properties | View other identity-true subgroup properties