Are the groups $G:=\{A \in M(n,\mathbb R) : A=A^t\}$ i.e. the group ( under addition ) of symmetric matrices and $O(n,\mathbb R)$ isomorphic?

Question

Let $G:=\{A \in M(n,\mathbb R) : A=A^t\}$ i.e. the group ( under addition ) of symmetric matrices ; Are $G$ and $O(n,\mathbb R)$ isomorphic ?

Answer 1

The answer is no, for every $n\ge1$. In $G$, if $A$ is not the identity element, so is not $A+A$. In $O(n,\mathbb R)$, however, there exists some $A$ such that $A^2$ is the identity element but $A$ isn't.

Answer 2

No, $G$ is abelian while $O(n,\Bbb R)$ is not.