Serial subgroup

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]

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This is a variation of normality|Find other variations of normality | Read a survey article on varying normality

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

• Ascendant subgroup
• Descendant subgroup
• Subnormal subgroup
• 2-subnormal subgroup
• Composition subgroup

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 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

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 other trim subgroup properties | View other trivially true subgroup properties | View other identity-true subgroup properties

The property of being a serial subgroup is trim, because the property of being normal is itself trim.