Algebraic group

In algebraic geometry, an algebraic group (or group variety) is a group that is an algebraic variety, such that the multiplication and inversion operations are given by regular maps on the variety.
In terms of category theory, an algebraic group is a group object in the category of algebraic varieties.
Classes.
Several important classes of groups are algebraic groups, including:
There are other algebraic groups, but Chevalley's structure theorem asserts that every algebraic group is an extension of an abelian variety by a linear algebraic group. More precisely, if "K" is a perfect field, and "G" an algebraic group over "K", there exists a unique normal closed subgroup "H" in "G", such that "H" is a linear algebraic group and "G"/"H" an abelian variety.
According to another basic theorem, any group that is also an affine variety has a faithful finite-dimensional linear representation: it is isomorphic to a matrix group, defined by polynomial equations.
Over the fields of real and complex numbers, every algebraic group is also a Lie group, but the converse is false
A group scheme is a generalization of an algebraic group that allows, in particular, working over a commutative ring instead of a field.
Algebraic subgroup.
An algebraic subgroup of an algebraic group is a Zariski-closed subgroup.
Generally these are taken to be connected (or irreducible as a variety) as well.
Another way of expressing the condition is as a subgroup that is also a subvariety.
This may also be generalized by allowing schemes in place of varieties. The main effect of this in practice, apart from allowing subgroups in which the connected component is of finite index > 1, is to admit non-reduced schemes, in characteristic "p".
Coxeter groups.
There are a number of analogous results between algebraic groups and Coxeter groups – for instance, the number of elements of the symmetric group is formula_1, and the number of elements of the general linear group over a finite field is the "q"-factorial formula_2; thus the symmetric group behaves as though it were a linear group over "the field with one element". This is formalized by the field with one element, which considers Coxeter groups to be simple algebraic groups over the field with one element.
Glossary of algebraic groups.
There are a number of mathematical notions to study and classify algebraic groups.
In the sequel, "G" denotes an algebraic group over a field "k".