Show that $H$ is a normal subgroup of $G$?

Question

Let $\mathbb M(n;\mathbb R)$ denote the set of all real matrices (identified with $\mathbb R^{n^2}$ and endowed with its usual topology) and $GL(n;\mathbb R)$ denote the group of all invertible matrices. Let $G$ be a subgroup of $GL(n;\mathbb R)$. Define $$H = \{ A \in G \mid \text{there exists } \phi : [0, 1] \rightarrow G \text{ continuous, such that } \phi (0)=A,\phi(1)=I\}.$$ Then is $H$ a normal subgroup?