# 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, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

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

If the ambient group is a finite group, this property is equivalent to the property:subnormal subgroup

View other properties finitarily equivalent to subnormal subgroup | View other variations of subnormal subgroup |

## Contents

## Definition

### Symbol-free definition

**PLACEHOLDER FOR INFORMATION TO BE FILLED IN**: [SHOW MORE]

### 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 in the whole group.ABOUT THIS PROPERTY: View variations of this property that are transitive | View variations of this property that are not transitiveABOUT 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.