3E: Exercises
- Page ID
- 132487
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Compute the following permutations:
- \((1435)(231)\)
- \(143)(231)^2\)
- \((1345)^{-1}(234)(123)\)
- \((1234)^4\)
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Consider the symmetric group \(S_4 \).
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Find all the subgroup lattice of \(S_4 \), the symmetric group of order 24.
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Find the centre of \(S_4 \).
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Find the normalizer of each subgroup of \(S_4 \).
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Find the cyclic subgroup of \(S_4 \) with an order of 4.
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Find a non-cyclic subgroup of \(S_4 \) with order 4.
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Consider the Dihedral group \(D_4 \).
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Find all the subgroup lattice of \(D_4 \), the Dihedral group of order 8.
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Find the centre of \(D_4 \).
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Find the normalizer of each subgroup \(D_4 \).
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Prove that every subgroup of \(D_4 \) of odd order is cyclic.
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Prove that the following groups are non-abelian.
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\(S_n \), for \(n\ge 3 \).
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\(A_n \) for \(n \ge 4 \).
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\(D_n \), for \(n \ge 3 \).
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Show that for \(n \ge 3 \), \(Z(S_n)=\{1\} \).
a. Find a cyclic subgroup of \(A_8 \) that has order 4.
b. Find a non-cyclic subgroup of \(A_8 \) that has order 4.
Suppose that \( \sigma \) and \( \tau \) in \(S_5 \) satisfying \( \tau \sigma = (1, 5, 2, 3) \) and \( \sigma \tau = (1, 2, 4, 5). \) If \( \sigma(1)=2, \) find \( \sigma \) and \( \tau. \)
1) Factor the permutation \((1,2,3,4,5)(6,7)(1,3,5,7)(1,6,3) \) into disjoint cycles.
2) Express the permutation \((1,2,3,4,5)(6,7)(1,3,5,7)(1,6,3) \) as products of transpositions, and decide if it is even or odd.
Let \( \sigma = (k_1, k_2, \cdots, k_r) \) be an \(r- \)cycle. If \( \gamma \in S_n, \) show that \( \gamma \sigma \gamma^{-1} \) is also a cycle; infact \( \gamma \sigma \gamma^{-1}= (\gamma(k_1), \gamma(k_2), \cdots, \gamma(k_r)) \)
Show that every product of \(3- \) cycles is an even permutation.
Show that every even permutation is a product of \(3- \) cycles.
Let \(G \) be a group. \(H=\{g^2: g\in G\}. \) Prove or disprove: \(H \) is a subgroup.
(Hint: \(G=A_4. \))
a. Find an element of maximum order in \(S_5\).
b. Find an element of maximum order in \(S_7\).
a. Let \(H\) be a subgroup of \(A_4\) containing \(K=\{e,(1,2)(3,4), (1,3)(2,4), (1,4)(2,3)\}\). Show that if \(H\) contains a \(3-\) cycle then \(H=A_4.\)
b. Show that \(A_4\) has no subgroup of order \(6.\)