# No common composition factor with quotient group is transitive

From Groupprops

This article gives the statement, and possibly proof, of a subgroup property (i.e., normal subgroup having no common composition factor with its quotient group) satisfying a subgroup metaproperty (i.e., transitive subgroup property)

View all subgroup metaproperty satisfactions | View all subgroup metaproperty dissatisfactions |Get help on looking up metaproperty (dis)satisfactions for subgroup properties

Get more facts about normal subgroup having no common composition factor with its quotient group |Get facts that use property satisfaction of normal subgroup having no common composition factor with its quotient group | Get facts that use property satisfaction of normal subgroup having no common composition factor with its quotient group|Get more facts about transitive subgroup property

## Contents

## Statement

Suppose is a group of finite composition length, and are subgroups such that is a normal subgroup of having no common composition factor with and is a normal subgroup of having no common composition factor with .

Then, is a normal subgroup of having no common composition factor with .

## Related facts

## Facts used

- No common composition factor with quotient group implies characteristic
- Characteristic of normal implies normal
- Third isomorphism theorem

## Proof=

**Given**: of finite composition length, with normal in and normal in , has no common composition factor with and has no common composition factor with .

**To prove**: is normal in and has no common composition factors with .

**Proof**:

- is normal in : This follows from facts (1) and (2).
- has no common composition factor with : Let be the set of composition factors of a group . Then, and . By assumption, and are disjoint. Also, since , and and are disjoint, and are disjoint. Thus, is disjoint from the union .