# Applications of the first isomorphism theorem

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.

• First Isomorphism theorem ... – DonAntonio Dec 26 '13 at 13:34

## 2 Answers

Hint:

The determinant map $\;\det: GL(n,\Bbb R)\to\Bbb R^*\;$ is a group homomorphism ...

Your second question is incomprehensible to me.

• I think he means that the naive description of $\mathbb{Z}/n\mathbb{Z}$ (as the numbers $\{0,1,2,\ldots, n-1\}$ under addition modulo $n$) actually coincides with the abstract quotient construction of $\mathbb{Z}/n\mathbb{Z}$ – Prahlad Vaidyanathan Dec 26 '13 at 13:42

If for the second one, you mean $(\mathbb Z/n\mathbb Z, +_{n\mathbb Z})$, then consider it as: $$\{0+n\mathbb Z,1+n\mathbb Z,\cdots,(n-1)+n\mathbb Z\}$$ and work with this map: $$\phi: \mathbb Z/n\mathbb Z\to\mathbb Z_n,~~(m+n\mathbb Z)=m,~~~~0\leq m<n$$ Show it is an group isomorphism.