Difference between revisions of "Center (algebra)"
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| − | In general, the '''center''' of an algebraic structure is the set of | + | In general, the '''center''' of an algebraic structure is the [[set]] of [[element]]s 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 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 == | == See also == | ||
| − | + | *[[Centralizer]] | |
| − | * [[Centralizer]] | ||
{{stub}} | {{stub}} | ||
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[[Category:Abstract algebra]] | [[Category:Abstract algebra]] | ||
Latest revision as of 01:48, 12 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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