# Right-quotient-transitively central factor implies join-transitively 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., right-quotient-transitively central factor) must also satisfy the second subgroup property (i.e., join-transitively central factor)

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## Contents

## Statement

### Verbal statement

Any right-quotient-transitively central factor is a join-transitively central factor.

### Statement with symbols

Suppose is a normal subgroup of a group such that for any subgroup of containing , such that is a central factor of , is also a central factor of .

Then, for any central factor of , the join of subgroups , which in this case is also the product of subgroups is also a central factor.

## Related facts

### Converse

## Facts used

## Proof

**Given**: is a normal subgroup of a group such that for any subgroup of containing , such that is a central factor of , is also a central factor of .

**To prove**: For any central factor of , the join of subgroups , which in this case is also the product of subgroups is also a central factor.

**Proof**: Let , and consider the quotient map . Then, .

- is a central factor of : By fact (1), since is a central factor of , is a central factor of , which translates to the above.
- is a central factor of : This follows from the assumption about in the given data.

This completes the proof.