Difference between revisions of "Center (group theory)"
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The '''center''' of a [[Group (mathematics)|group]] <math>G</math> (denoted <math>Z(G)</math>) is the set of all elements which [[Commutativity|commute]] with every element of the group. That is, the center of <math>G</math> is <math>\{a \in G : ab=ba\ \forall b \in G\}</math>. | The '''center''' of a [[Group (mathematics)|group]] <math>G</math> (denoted <math>Z(G)</math>) is the set of all elements which [[Commutativity|commute]] with every element of the group. That is, the center of <math>G</math> is <math>\{a \in G : ab=ba\ \forall b \in G\}</math>. | ||
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| + | ==Facts== | ||
| + | *If <math>G</math> is a group, the center <math>Z(G)</math> is a [[normal subgroup]] of <math>G</math>. | ||
| + | *We have <math>Z(G) = G</math> if and only if <math>G</math> is an [[abelian group]]. | ||
==External Links== | ==External Links== | ||
*[http://mathworld.wolfram.com/GroupCenter.html Center] on MathWorld | *[http://mathworld.wolfram.com/GroupCenter.html Center] on MathWorld | ||
[[Category:Algebra]] | [[Category:Algebra]] | ||
Revision as of 03:39, April 10, 2013
The center of a group (denoted ) is the set of all elements which commute with every element of the group. That is, the center of is .
Facts
- If is a group, the center is a normal subgroup of .
- We have if and only if is an abelian group.
External Links
- Center on MathWorld