Let $G$ be an algebraic group, and $G\_{Id}$ the connected component of the identity. Then $G\_{Id}$ is a normal subgroup of $G$ and $G/G\_{Id}$ is the component group of $G$.


Let $G\_{c}\subset G$ be another connected component of $G$. Is is possible to define a group structure on $G\_c$?


Assume we know that for any connected component $G\_c$ of $G$ we have an injective morphism $f\_c:G\_c\rightarrow H\times F$ where $H$ is a group and $F$ a finite group. Now, for any $g\in g$ there exists a unique connected component $G\_c$ such that $g\in G\_c$ and we may define a map 
$$f:G\rightarrow H\times F,\: g\mapsto f\_c(g).$$
If $f$ is surjective can we conclude that $f$ is an isomorphism?



Connected components of algebraic groups
Each connected component is (in a natural way) a torsor under the identity component. The choice of a rational point (if there is one) defines an isomorphism with the identity component, and makes the component into an algebraic group.


