Structure theorem for general algebraic groups

Statement

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, then 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.