# Central factor is not quotient-transitive

From Groupprops

This article gives the statement, and possibly proof, of a subgroup property (i.e., central factor)notsatisfying a subgroup metaproperty (i.e., quotient-transitive subgroup property).

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## Contents

## Statement

It is possible to have groups such that:

- is a central factor of . (In particular, is normal in ).
- is a central factor of .
- is not a central factor of .

## Definitions used

### Central factor

`Further information: Central factor`

A subgroup of a group is termed a **central factor** if where is the centralizer of in .

Note that any central subgroup, i.e., any subgroup contained in the center, is a central factor.

## Related facts

- Centrality is not quotient-transitive
- Direct factor is quotient-transitive
- Central factor over direct factor implies central factor

## Facts used

## Proof

### A generic example

Let be a finite non-Abelian group of nilpotence class two and be maximal among Abelian normal subgroups of . Consider . Then:

- is a central factor of : In fact, is a central subgroup of -- it is contained in the center of .
- is a central factor of : Since is normal in , we have , and since has class two, we have . Thus, . Thus, is trivial, so is central in . Thus, is a central factor of .
- is not a central factor of : By fact (1), is a self-centralizing subgroup of . Since is proper, this yields that is
*not*a central factor of .

### Example of the dihedral group

`Further information: dihedral group:D8`

Consider the dihedral group:

Define:

.

Then, we have:

- is a central factor of : In fact, equals the center of .
- is a central factor of : In fact, is Abelian, so any subgroup of it is a central factor.
- is not a central factor of : is a proper self-centralizing subgroup of , so is not a central factor of .

### Example of the quaternion group

`Further information: quaternion group`

Consider the quaternion group, with identity element , and with the list of elements:

.

Define:

.

Then, we have:

- is a central factor of : In fact, equals the center of .
- is a central factor of : In fact, is Abelian, so any subgroup of it is a central factor.
- is not a central factor of : is a proper self-centralizing subgroup of , so is not a central factor of .