4.3: Exercises
- Page ID
- 84812
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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.
- Every group contains at least two distinct subgroups.
- If \(H\) is a proper subgroup of group \(G\) and \(G\) is finite, then we must have \(|H|\lt |G|\text{.}\)
- \(7\mathbb{Z}\) is a subgroup of \(14\mathbb{Z}\text{.}\)
- A group \(G\) may have two distinct proper subgroups which are isomorphic (to one another).
2. Give specific, precise examples of the following groups \(G\) with subgroups \(H\text{:}\)
- A group \(G\) with a proper subgroup \(H\) of \(G\) such that \(|H|=|G|\text{.}\)
- A group \(G\) of order \(12\) containing a subgroup \(H\) with \(|H|=3\text{.}\)
- A nonabelian group \(G\) containing a nontrivial abelian subgroup \(H\text{.}\)
- A finite subgroup \(H\) of an infinite group \(G\text{.}\)
3. Let \(n\in \mathbb{Z}^+\text{.}\)
- Prove that \(n\mathbb{Z} \leq \mathbb{Z}\text{.}\)
- Prove that the set \(H=\{A\in \mathbb{M}_n(\mathbb{R})\,:\,\det A=\pm 1\}\) is a subgroup of \(GL(n,\mathbb{R})\text{.}\)
(Note: Your proofs do not need to be long to be correct!)
4. Let \(n\in \mathbb{Z}^+\text{.}\) For each group \(G\) and subset \(H\text{,}\) decide whether or not \(H\) is a subgroup of \(G\text{.}\) In the cases in which \(H\) is not a subgroup of \(G\text{,}\) provide a proof. (Note. Your proofs do not need to be long to be correct!)
- \(G=\mathbb{R}\text{,}\) \(H=\mathbb{Z}\)
- \(G=\mathbb{Z}_{15}\text{,}\) \(H=\{0,5,10\}\)
- \(G=\mathbb{Z}_{15}\text{,}\) \(H=\{0,4,8,12\}\)
- \(G=\mathbb{C}\text{,}\) \(H=\mathbb{R}^*\)
- \(G=\mathbb{C}^*\text{,}\) \(H=\{1,i,-1,-i\}\)
- \(G=\mathbb{M}_n(\mathbb{R})\text{,}\) \(H=GL(n,\mathbb{R})\)
- \(G=GL(n,\mathbb{R})\text{,}\) \(H=\{A\in \mathbb{M}_n(\mathbb{R})\,:\,\det A = -1\}\)
5. Let \(G\) and \(G'\) be groups, let \(\phi\) be a homomorphism from \(G\) to \(G'\text{,}\) and let \(H\) be a subgroup of \(G\text{.}\) Prove that \(\phi(H)\) is a subgroup of \(G'\text{.}\)
6. Let \(G\) be an abelian group, and let \(U=\{g\in G\,:\, g^{-1}=g\}.\) Prove that \(U\) is a subgroup of \(G\text{.}\)