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\(\def\Z{\mathbb{Z}} \def\zn{\mathbb{Z}_n} \def\znc{\mathbb{Z}_n^\times} \def\R{\mathbb{R}} \def\Q{\mathbb{Q}} \def\C{\mathbb{C}} \def\N{\mathbb{N}} \def\M{\mathbb{M}} \def\G{\mathcal{G}} \def\0{\mathbf 0} \def\Gdot{\langle G, \cdot\,\rangle} \def\phibar{\overline{\phi}} \DeclareMathOperator{\lcm}{lcm} \DeclareMathOperator{\Ker}{Ker} \def\siml{\sim_L} \def\simr{\sim_R} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \)

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 \(G\simeq G'\text{.}\)