Rank of an algebraic group: Difference between revisions
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The '''rank of an algebraic group''' is defined as the [[defining ingredient::dimension of an algebraic group|dimension]] of any [[defining ingredient::Cartan subgroup]] of it. | The '''rank of an algebraic group''' is defined as the [[defining ingredient::dimension of an algebraic group|dimension]] of any [[defining ingredient::Cartan subgroup]] of it. | ||
For a [[finite-dimensional algebraic group]], the rank is also finite and is at most equal to the dimension. Equality holds if and only if the group is (up to subgroups of finite index) | For a [[finite-dimensional algebraic group]], the rank is also finite and is at most equal to the dimension. Equality holds if and only if the group is (up to subgroups of finite index) [[nilpotent algebraic group|nilpotent]]. | ||
Latest revision as of 18:09, 1 January 2012
Definition
The rank of an algebraic group is defined as the dimension of any Cartan subgroup of it.
For a finite-dimensional algebraic group, the rank is also finite and is at most equal to the dimension. Equality holds if and only if the group is (up to subgroups of finite index) nilpotent.