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Serial subgroup
From Groupprops
This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof.
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RANDOM TIP:The testing section provides information on practical testing for the subgroup property, including implementation in GAP, when possible.
This is a variation of normality
View a complete list of variations of normality OR read a survey article on varying normality
Contents |
Definition
Symbol-free definition
A subgroup of a group is termed a serial subgroup if there is a normal series starting at the subgroup and ending at the whole group.
In terms of the serial closure operator
The property of being a serial subgroup is obtained by applying the serial closure operator to the subgroup property of being normal.
Relation with other properties
Stronger properties
Related group properties
- Absolutely simple group is a group that has no proper nontrivial serial subgroup. Thus, this property is obtained by applying the simple group operator to the subgroup property of being simple
Metaproperties
Transitivity
This subgroup property is transitive: a subgroup with this property in a subgroup with this property, also has this property.
View a complete list of transitive subgroup properties|View a complete list of facts related to transitivity of subgroup properties
For full proof, refer: Serial satisfies transitivity
Since any serially closed subgroup property is transitive, the property of being a serial subgroup is a transitive subgroup property.
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 all trim subgroup properties OR view trivially true subgroup properties OR view identity-true subgroup properties
The property of being a serial subgroup is trim, because the property of being normal is itself trim.
