2E: Exercises
- Page ID
- 131663
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Check the following structures for the group properties:
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\(\mathbb{Z}_4 \) under addition.
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\(\mathbb{Z}_5^* \) under multiplication.
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\( \mathbb{Z}_8^* \) under multiplication.
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\(\{ \pm 1, \pm i\} \) under multiplication.
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Rotations that map a brick to itself.
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The four functions \(f(x)=x, g(x)=-x, h(x)= \dfrac{1}{x}, j(x)= \dfrac{-1}{x}\) under composition.
-
The four matrices \[ \begin{bmatrix}
1 & 0 \\
0 & 1
\end{bmatrix},
\begin{bmatrix}
-1 & 0 \\
0 & -1
\end{bmatrix},
\begin{bmatrix}
1 & 0 \\
0 & -1
\end{bmatrix},
\begin{bmatrix}
-1 & 0 \\
0 & 1
\end{bmatrix}
\]
under multiplication.
Let
\(H(C) =\Bigg\{ \begin{bmatrix}1 & a & b\\
0 & 1 & c \\
0 & 0 & 1 \end{bmatrix}\Big| \ a,b,c, \in \mathbb{C}\Bigg\}\).
Show that \(H(\mathbb{C})\) is a group under matrix multiplication. Demonstrate explicitly that \(H(\mathbb{C})\) is always non-abelian.
Let \(S = \mathbb{R} \backslash \{-1\}\) and define a binary operation \(\oplus\) on \(S\) by \(a \oplus b = a+b+ab\). Prove that \((S, \oplus)\) is an abelian group.
Given the groups \(\mathbb{R}^*\) and \(\mathbb{Z}\), let \(G = \mathbb{R}^* \times \mathbb{Z}\). Define a binary operation \(\star\) by \((a,m) \star (b, n) = (ab,m + n)\). Show that \( (G, \star)\) is a group under this operation.
Let \(G\) be a group. Show that if \(a^2 = e\), for all elements of \(a \in G\), then \(G\) must be abelian.
Let \(H\) consists of \(2 \times 2\) matrices of the form
\(\begin{bmatrix} \cos{(x)} & -\sin{(x)}\\
\sin{(x)} & \cos{(x)} \end{bmatrix}\), where \(x \in \mathbb{R}\). Prove that \(H\) is a subgroup of \(SL_2(\mathbb{R})\).
Prove or disprove the following statements: Let \(H\) and \(K\) be subgroups of a group \(G\).
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\(H \cup K\) is a subgroup of \(G\).
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\(H \cap K\) is a subgroup of \(G\).
Prove that for each element \(a \in G\), where \(G\) is a group, the centralizer of \(a\), \(C(a)\) is a subgroup of \(G\). Prove that for each element \(a \in G\), where \(G\) is a group, that \(C(a)=C(a^{-1})\).
Let \(G=GL(2,\mathbb{R})\). Then find
-
\(C \Bigg(\begin{bmatrix}1 & 1\\
1 & 0 \end{bmatrix}\Bigg)\). -
\(C \Bigg( \begin{bmatrix} 0 & 1\\
1 & 0 \end{bmatrix}\Bigg)\). -
\(\mathbb{Z}(G)\).
Let \(G\) be a group. \(H=\{g^2: g\in G\}. \)
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Prove or disprove: If \(G\) is abelian then \(H\) is a subgroup.
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Prove or disprove: If \(H\) is a subgroup then \(G\) is abelian.
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List the cyclic subgroups of \(U(30) \).
\(U(30)=\{1,7,11,13,17,19,23,29\} \).
Thus \(|U(30)|=8 \).
Show that \(U(20)\ne <k> \) for any \(k \) in \(U(20) \). [Hence, \(U(20) \) is not cyclic.]
\(U(20)=\{1,3,7,9,11,13,17,19\} \).
\(|U(20)|=8 \).
Decide whether \(U(10) \) is cyclic or not.
Is \((\mathbb{Z},+) \) cyclic group? If so, what are the possible generators?