# 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 $G$ be a group. The center of $G$, denoted $Z(G)$, is defined as follows:

$Z(G) := \{ g \in G \mid gh = hg \ \forall \ h \in G \}$.

In other words, $Z(G)$ is the set of those elements of $G$ that commute with every element of $G$.

### Quasiautomorphism

Further information: Quasihomomorphism of groups, Quasiautomorphism

Let $G$ and $H$ be groups. A function $\varphi:G \to H$ is termed a quasihomomorphism of groups if whenever $a,b \in G$ commute, we have $\varphi(ab) = \varphi(a)\varphi(b)$.

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.