9.3: Exercises
1. Let \(F\) be the group of all functions from \([0,1]\) to \(\mathbb{R}\text{,}\) 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 \(\mathbb{R}\text{.}\)
2. Let \(N=\{1,-1\}\subseteq \mathbb{R}^*\text{.}\) Prove that \(\mathbb{R}^*/N\) is a group that's isomorphic to \(\mathbb{R}^+\text{.}\)
3. Let \(n\in \mathbb{Z}^+\) and let \(H=\{A\in GL(n,\mathbb{R})\,:\, \det A =\pm 1\}\text{.}\) Identify a group familiar to us that is isomorphic to \(GL(n,\mathbb{R})/H\text{.}\)
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 \(G\simeq G'\text{.}\)