---
tag: H7VXZAK6VN
kind: statement
role: theorem
name: Structure theorem for general algebraic groups
sources:
  - wikipedia: en:247924@r1373258378
    locator: Algebraic group, section Structure theorem for general algebraic groups
    note: adopted
    mention: wikipedia://en/247924#K297
    sections: [Statement]
inventory: [QWBN7CPYF7]
---

## 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.