Center is marginal of finite type

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., marginal subgroup of finite type)}
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Statement

The center of a group is a marginal subgroup of finite type, and hence a marginal subgroup. More specifically, it is the marginal subgroup corresponding to the commutator word $w(x_1,x_2) = x_1x_2x_1^{-1}x_2^{-1}$.

Related facts

Corollaries

Property	Statement about marginal subgroups	Corresponding statement about center
bound-word subgroup	marginal implies bound-word	center is bound-word
strictly characteristic subgroup	marginal implies strictly characteristic	center is strictly characteristic
finite direct power-closed characteristic subgroup	marginal implies direct power-closed characteristic	center is direct power-closed characteristic
characteristic subgroup	marginal implies characteristic	center is characteristic