identification of $GL_n(\mathbb{R})/H$

Question

could any one just tell me can I identify the group $GL_n(\mathbb{R})/H$ with the group $(\mathbb{R}^{+},.)$? where $H=$ Normal subgroup of matrices with positive determinant. .Any correct answers and hint will be appreciated, what I did was just a gues, I dont know how what will be the identification and how.Thank you

Answer 1

If $n$ is even, then for every $M \in GL_n(\mathbb{R})$, either $M \in H$ or $M=-M'$ with $M'=-M \in H$ (because $\det(-M)=(-1)^n\det(M)=- \det(M)$). So $GL_n(\mathbb{R})/H \simeq \mathbb{Z}_2$.

However, $M \mapsto |\det(M)|$ is an epimorphism from $GL_n(\mathbb{R})$ to $\mathbb{R}_+$ and its kernel is $SL_n(\mathbb{R})= \{ M \in GL_n(\mathbb{R}) : \det(M)=\pm 1\}$, so $GL_n(\mathbb{R})/SL_n(\mathbb{R}) \simeq \mathbb{R}_+$.