# Central subgroup

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In mathematics, in the field of group theory, a subgroup of a group is termed **central** if it lies inside the center of the group.

Given a group , the center of , denoted as , is defined as the set of those elements of the group which commute with every element of the group. The center is a characteristic subgroup and is also an abelian group (because, in particular, all elements of the center must commute with each other). A subgroup of is termed *central* if .

Central subgroups have the following properties:

- They are abelian groups.
- They are normal subgroups. In fact, they are central factors, and are hence transitively normal subgroups.

## References

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