# Central factor implies transitively normal

From Groupprops

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., central factor) must also satisfy the second subgroup property (i.e., transitively normal subgroup)

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## Statement

### Verbal statement

Any central factor of a group is a transitively normal subgroup. In other words, any normal subgroup of a central factor is a central factor.

### Statement with symbols

Suppose is a central factor of a group . Then, is a transitively normal subgroup of , i.e., if is a normal subgroup of , then is a normal subgroup of .

## Proof

**Given**: A group , a central factor of , a normal subgroup of . is an element of and is an element of .

**To prove**: .

**Proof**: Let be the inner automorphism obtained as conjugation by .

- There exists such that the inner automorphism by , as an inner automorphism of , is the restriction to of . In other words, for any , : This follows from the definition of central factor.
- : This follows from the fact that is normal in .
- : This follows immediately from the previous two steps.