Suppose you have a group isomorphism given by the first isomorphism theorem:
$G/ker(\phi) \simeq im(\phi)$
What can we say about the group $ker(\phi)\times im(\phi)$? In particular, when does the following hold:
$G\simeq ker(\phi)\times im(\phi)$?
I ask this question because i want to prove that $GL_n^+(\mathbb{R}) \simeq SL_n(\mathbb{R}) \times \mathbb{R}^*_{>0}$, with $GL_n^+(\mathbb{R})$ the group of matrices with positive determinant. I proved that $SL_n(\mathbb{R})$ is a normal subgroup and that $GL_n^+(\mathbb{R})/ SL_n(\mathbb{R}) \simeq \mathbb{R}^*_{>0}$, using the surjective homomorphism $det(M)$. I tried something with semidirect products but I got stuck.