*Z*(*G*) is a subgroup of *G*: in fact, if *x* and
*y* are in *Z*(*G*), then for each *g* in *G*

*(xy)g*=*x(yg)*=*x(gy)*=*(xg)y*=*(gx)y*=*g(xy)*

Moreover, *Z*(*G*) is an abelian subgroup of *G*, and a normal subgroup of *G*, and even a
characteristic subgroup of it.

See also center (algebra).