Is the group of all determinants of all invertible $n \times n$-matrices isomorphic to $\langle\mathbb{R}^*, \cdot\rangle$? I am doing Linear Algebra and Abstract Algebra simultaneously, and in Linear Algebra class, going through determinants, I thought of something interesting (for a freshman just learning the subjects, that is):
Is the group of all determinants of all invertible matrices isomorphic to the non-zero real numbers under multiplication? While I expect the answer to be yes, there may be some details that I have missed/misunderstood.

Yes. Consider the diagonal matrix with $r, 1, 1, ..., 1$ on the diagonal, where $r$ is a nonzero real. That has determinant $r$ so "det" is a surjective map from $GL(n, \mathbb R)$ to $ \mathbb R^{*}$. And since $\det(AB) = \det(A) \det(B)$, it's actually a homomorphism. 
