# Center is marginal of finite type

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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