groups are affine algebraic groups. unipotent groups. algebraic tori. certain semidirect products, for instance Jet groups, or some solvable groups such as that of invertible triangular matrices. Linear algebraic groups can be classified to a certain extent. Levi's theorem states that every such is (essentially) a semidirect product of a unipotent group (its unipotent radical) with a reductive group. In turn reductive groups are decomposed as (again essentially) a product of their center (an algebraic torus) with a semisimple group. The latter are classified over algebraically closed fields via their Lie algebra. The classification over arbitrary fields is more involved but still well-understood. If can be made very explicit in some cases, for example over the real or p-adic fields, and thereby over number fields via local-global principles. Abelian varieties Abelian varieties are connected projective algebraic groups, for instance elliptic curves. They are always commutative. They arise naturally in various situations in algebraic geometry and number theory, for example as the Jacobian variety of a curve. Structure theorem for general algebraic groups Not all algebraic groups are linear groups or abelian varieties, for instance some group schemes occurring naturally in arithmetic geometry are neither. Chevalley's structure theorem asserts that every connected algebraic group is an extension of an abelian variety by a linear algebraic group. More precisely, if K is a perfect field, and G a connected algebraic group over K, there exists a unique normal closed subgroup H in G, such that H is a connected linear algebraic group and G/H an abelian variety. Connectedness As an algebraic variety carries a Zariski topology. It is not in general a group topology, i.e. the group operations may not be continuous for this topology (because Zariski topology on the product is not the product of Zariski topologies on the factors). An algebraic group is said to be connected if the underlying algebraic variety is connected for the Zariski topology. For an algebraic group this means that it is not the union of two proper algebraic subsets. Examples of groups which are not connected are given by the algebraic subgroup of th roots of unity in the multiplicative group (each point is a Zariski-closed subset so it is not connected for ). This group is generally denoted by . Another non-connected group are orthogonal group in even dimension (the determinant gives a surjective morphism to ). More generally every finite group is