Prop: If $G$ is a Matrix Lie group, then the connected component that contains the identity $I$ is a normal subgroup of $G$.
I have problem in the proof of this. Suppose that $A$ and $B$ belong to the connected component that contains $I$, then exist two continuos function ($A(t), B(t)$) in $G$ such that $A(0)=B(0)=I$ and $A(1)=A$, $B(1)=B$.
My question is: if we consider $A(t)B(t)$ why this path is contained in the connected component of $G$?
Best Answer
Because the mapt $t\mapsto A(t)B(t)$ is continuous and its domain is conneced (it is an interval). Therefore, its range is connected too. Since, furtheremore, $\operatorname{Id}$ belongs to the range, the range is a subset of the connected component of $\operatorname{Id}$.