3.4: Exercises
- Page ID
- 84808
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- True/False. For each of the following, write T if the statement is true; otherwise, write F. You do NOT need to provide explanations or show work for this problem. Throughout, let \(G\) and \(G'\) be groups.
- If there exists a homomorphism \(\phi\,:\,G\to G'\text{,}\) then \(G\) and \(G'\) must be isomorphic groups.
- There is an integer \(n\geq 2\) such that \(\mathbb{Z}\simeq \mathbb{Z}_n\text{.}\)
- If \(|G|=|G'|=3\text{,}\) then we must have \(G\simeq G'\text{.}\)
- If \(|G|=|G'|=4\text{,}\) then we must have \(G\simeq G'\text{.}\)
- For each of the following functions, prove or disprove that the function is (i) a homomorphism; (ii) an isomorphism. (Remember to work with the default operation on each of these groups!)
- The function \(f:\mathbb{Z}\to\mathbb{Z}\) defined by \(f(n)=2n\text{.}\)
- The function \(g:\mathbb{R}\to\mathbb{R}\) defined by \(g(x)=x^2\text{.}\)
- The function \(h:\mathbb{Q}^*\to\mathbb{Q}^*\) defined by \(h(x)=x^2\text{.}\)
- Define \(d : GL(2,\mathbb{R})\to \mathbb{R}^*\) by \(d(A)=\det A\text{.}\) Prove/disprove that \(d\) is:
- a homomorphism
- 1-1
- onto
- an isomorphism.
- Complete the group tables for \(\mathbb{Z}_4\) and \(\mathbb{Z}_8^{\times}\text{.}\) Use the group tables to decide whether or not \(\mathbb{Z}_4\) and \(\mathbb{Z}_8^{\times}\) are isomorphic to one another. (You do not need to provide a proof.)
- Let \(n\in \mathbb{Z}^+\text{.}\) Prove that \(\langle n\mathbb{Z},+\rangle \simeq \langle \mathbb{Z},+\rangle\text{.}\)
- Let \(G\) and \(G'\) be groups, where \(G\) is abelian and \(G\simeq G'\text{.}\) Prove that \(G'\) is abelian.
- Give an example of groups \(G\) and \(G'\text{,}\) where \(G\) is abelian and there exists a homomorphism from \(G\) to \(G'\text{,}\) but \(G'\) is NOT abelian.
- Let \(\langle G,\cdot\rangle\) and \(\langle G',\cdot'\rangle\) be groups with identity elements \(e\) and \(e'\text{,}\) respectively, and let \(\phi\) be a homomorphism from \(G\) to \(G'\text{.}\) Let \(a\in G\text{.}\) Prove that \(\phi(a)^{-1}=\phi(a^{-1})\text{.}\)