# Direct factor implies central factor

This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., direct factor) must also satisfy the second subgroup property (i.e., central factor)

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## Contents

## Statement

### Verbal statement

Any direct factor of a group is a central factor.

### Statement with symbols

Suppose is a direct factor of a group , i.e., is a normal subgroup of and there exists a normal subgroup of such that and is trivial. Then, is a central factor of , i.e., .

## Related facts

### Converse

- Central factor not implies direct factor
- Central factor not implies complemented
- Complemented central factor not implies direct factor

### Stronger facts

- Direct factor implies right-quotient-transitively central factor
- Direct factor implies join-transitively central factor
- Join of finitely many direct factors implies central factor

- Image-potentially direct factor equals central factor
- Internal direct product implies internal central product
- Direct factor implies transitively normal
- Central factor implies transitively normal

## Facts used

## Proof

### Proof using the product with centralizer definition

**Given**: A group , normal subgroups of such that and is trivial.

**To prove**: .

**Proof**:

- Every element of commutes with every element of : For and , the commutator is in (because is normal) and is also in (because is normal). (This is based on one of the equivalent definitions of normal subgroup. It can also be seen by seeing that ). Since is trivial, we obtain that is the identity element, so .
- : This is a reformulation of the previous step.
- : Since , . Equality holds throughout, so .

Note that step (1) above is sometimes taken as part of the definition of internal direct product, in which case it does not need to be proved.

### Hands-off proof using direct product and central product

By fact (1), an internal direct product is an internal central product. A direct factor is a factor in an internal direct product, and a central factor is a factor in an internal central product. Thus, a direct factor must be a central factor.

### Proof using the inner automorphism definition of central factor

**Given**:A group that is an internal direct product of subgroups and . In other words, and is trivial.

**To prove**: For any , there exists such that, restricted to , conjugation by equals conjugation by .

**Proof**: We can write with , since .

- Every element of commutes with every element of : For and , the commutator is in (because is normal) and is also in (because is normal). (This is based on one of the equivalent definitions of normal subgroup. It can also be seen by seeing that ). Since is trivial, we obtain that is the identity element, so .
- Restricted to , conjugation by equals conjugation by : If denote conjugation by respectively, then ,math>c_g = c_a \circ c_b</math>. Restricting to , is trivial by the previous step. Thus, .

### Proof using the inner automorphism definition and identifying with the external direct product

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