Difference between revisions of "Center (algebra)"
(New page: In general, the '''center''' of an algebraic structure is the set of elements which commute with every of the structure. With groups, this definition is straightforward; for rings...) |
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| − | In general, the '''center''' of an algebraic structure is the set of elements which commute with every of the structure. With [[group]]s, this definition is straightforward; for [[ring]]s and [[field]]s, the commutativity in question is multiplicative commutativity. | + | In general, the '''center''' of an algebraic structure is the set of elements which commute with every of the structure. With [[magma]]s (such as [[group]]s), this definition is straightforward; for [[ring]]s and [[field]]s, the commutativity in question is multiplicative commutativity. |
The center of a group is never empty, as the identity commutes with every element of a group. The center of a group is a [[subgroup]] of the group—a [[normal subgroup]], in fact; it is also stable under any [[endomorphism]] on the group. | The center of a group is never empty, as the identity commutes with every element of a group. The center of a group is a [[subgroup]] of the group—a [[normal subgroup]], in fact; it is also stable under any [[endomorphism]] on the group. | ||
Revision as of 11:29, 11 May 2008
In general, the center of an algebraic structure is the set of elements which commute with every of the structure. With magmas (such as groups), this definition is straightforward; for rings and fields, the commutativity in question is multiplicative commutativity.
The center of a group is never empty, as the identity commutes with every element of a group. The center of a group is a subgroup of the group—a normal subgroup, in fact; it is also stable under any endomorphism on the group.
See also
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