# Group in which every normal subgroup is a central factor

From Groupprops

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 propertiesVIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

## Contents

## Definition

A **group in which every normal subgroup is a central factor** is a group satisfying the following equivalent conditions:

- Every normal subgroup is a central factor: In other words, the product of any normal subgroup and its centralizer is the whole group.
- Every subnormal subgroup is a central factor: In other words, the product of any subnormal subgroup and its centralizer is the whole group.

## Formalisms

### 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 (normal subgroup) satisfies the second property (central factor), and vice versa.

View other group properties obtained in this way