I would like to know if possible how to use the first isomorphism theorem to solve the following problems:
(1) Consider the general linear group $GL(n,R)$. Show that the set of all $n\times n$ matrices with determinant $1$ is a normal subgroup of $GL(n,R)$
(2) Show that the groups $(Z⁄nZ,+nZ)$ and $(Z_n,+n)$ are isomorphic. I know the first isomorphism theorem states that: Let $φ\colon G\to G'$ be a group homomorphism and denote $\ker(φ)$ by $H$. Then $H$ is a normal subgroup of $G$ and the map $f\colon G⁄H\to φ[G]$ given by $f(gH)= φ(g)$ is an isomorphism.