# Center is normal

From Groupprops

This article gives the statement, and possibly proof, of the fact that for any group, the subgroup obtained by applying a given subgroup-defining function (i.e., center) always satisfies a particular subgroup property (i.e., normal subgroup)}

View subgroup property satisfactions for subgroup-defining functions View subgroup property dissatisfactions for subgroup-defining functions

## Contents

## Statement

The center of any group is a normal subgroup.

## Related facts

### More about the center

- The quotient group of the group by its center is isomorphic to the inner automorphism group. This is because the center is the kernel of the group action on itself by conjugation, whose image is the inner automorphism group.

### Stronger subgroup properties satisfied

- Hereditarily normal subgroup:
*Every*subgroup of the center is a normal subgroup.`For full proof, refer: center is hereditarily normal` - Central subgroup, central factor, transitively normal subgroup
- Characteristic subgroup:
`For full proof, refer: center is characteristic` - Strictly characteristic subgroup:
`For full proof, refer: center is strictly characteristic` - Bound-word subgroup:
`For full proof, refer: center is bound-word`

### Stronger subgroup properties not satisfied

- Fully invariant subgroup:
`For full proof, refer: center not is fully invariant` - Injective endomorphism-invariant subgroup:
`For full proof, refer: center not is injective endomorphism-invariant` - Weakly normal-homomorph-containing subgroup:
`For full proof, refer: center not is weakly normal-homomorph-containing`