Let $GL_{n}(\mathbb{R})=\{A_{n\times n}\mid A \text{ invertible matrice}\}$, $SL_{n}(\mathbb{R})=\{A\in GL_{n}(\mathbb{R})\mid det(A)=1\}$ be a special linear group and $SO(n)=\{A\in O(n)\mid det(A)=1\}$ be a special orthogonal group.
Theorem: Let $A\in SL_{n}(\mathbb{R})$. Then there exists a rotation matrix $R\in SO(n)$ and a real, symmetric and positive semidefinite matrix $P\in SL_{n}(\mathbb{R})$ such that $A=RP$.
By this theorem, $SL_{n}(\mathbb{R})$ is divisible $GL_{n}(\mathbb{R})$-module and $SO(n)$ is divisible $GL_{n}(\mathbb{R})$-module, but $O(n)$, is not divisible $GL_{n}(\mathbb{R})$-module. I want to show that $O(n)$ is not injective $GL_{n}(\mathbb{R})$-module, by choosing $g\colon K\to O(n)$, such that $g\neq hf$ for every monomorphism $f\colon K\to GL_{n}(\mathbb{R})$ and homomorphism $h\colon GL_{n}(\mathbb{R})\to O(n)$, but I don't know where to start. Thank you in advance.