# Center is quasiautomorphism-invariant

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., quasiautomorphism-invariant subgroup)}
View subgroup property satisfactions for subgroup-defining functions  View subgroup property dissatisfactions for subgroup-defining functions

## Statement

The center of a group is a quasiautomorphism-invariant subgroup: it is invariant under all quasiautomorphisms of the group.

## Definitions used

### Center

Further information: Center

Let  be a group. The center of , denoted , is defined as follows:

.

In other words,  is the set of those elements of  that commute with every element of .

### Quasiautomorphism

Further information: Quasihomomorphism of groups, Quasiautomorphism

Let  and  be groups. A function  is termed a quasihomomorphism of groups if whenever  commute, we have .

A function from a group to itself is termed a quasiautomorphism if it is a quasihomomorphism and has a two-sided inverse that is also a quasihomomorphism.

### Quasiautomorphism-invariant subgroup

Further information: Quasiautomorphism-invariant subgroup

A subgroup of a group is termed quasiautomorphism-invariant if for every quasiautomorphism of the group, the subgroup gets mapped to within itself.