# Section9.3Exercises

##### 1

Let $F$ be the group of all functions from $[0,1]$ to $\R\text{,}$ under pointwise addition. Let

\begin{equation*} N=\{f\in F: f(1/4)=0\}. \end{equation*}

Prove that $F/N$ is a group that's isomorphic to $\R\text{.}$

##### 2

Let $N=\{1,-1\}\subseteq \R^*\text{.}$ Prove that $\R^*/N$ is a group that's isomorphic to $\R^+\text{.}$

##### 3

Let $n\in \Z^+$ and let $H=\{A\in GL(n,\R)\,:\, \det A =\pm 1\}\text{.}$ Identify a group familiar to us that is isomorphic to $GL(n,\R)/H\text{.}$

##### 4

Let $G$ and $G'$ be groups with respective normal subgroups $N$ and $N'\text{.}$ Prove or disprove: If $G/N\simeq G'/N'$ then $G\simeq G'\text{.}$