H = {A $\in$ G| there exists f:[0,1]$\to$G continuous, such that f(0)=A, f(1)=I}, Is H normal in G?

Question

If G is a subgroup of GL(n;$\mathbb R$) and H = {A $\in$ G| there exists f:[0,1]$\to$G continuous, such that f(0)=A, f(1)=I}, Is H normal in G?

Answer 1

Hint Component of Identity is a Normal Subgroup.

Answer 2

To show $B^{-1} A B \in H$, consider the function $g(t) = B^{-1} f(t) B$.