Toroidal Affine Nash Groups

A toroidal affine Nash group is the affine Nash group analogue of an anti-affine algebraic group.In this note, we prove analogues of Rosenlicht's structure and decomposition theorems: (1) Every affine Nash group G has a smallest normal affine Nash subgroup H such that G/H is an almost linear affine Nash group, and this H is toroidal.(2) If G is a connected affine Nash group, then there exist a largest toroidal affine Nash subgroup G ant and a largest connected, normal, almost linear affine Nash subgroup G aff .Moreover, we have G = G ant G aff , and G ant G aff contains (G ant ) aff as an affine Nash subgroup of finite index.