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.

Related facts

Related subgroup properties satisfied by the center

Related subgroup properties not satisfied by the center