Test 2
- Page ID
- 167477
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Consider the symmetric group \(S_7\).
- Factor the permutation \((1 2 3 4 5)(67)(1357)(613)\) into disjoint cycles.
- Identify all possible types of permutations and specify the quantity of each type.
- Find the possible order of all the elements of \(S_7\).
- Show that \(S_7\) is not abelian.
Consider the group \(\mathbb{Z}_{12}\) with addition modulo \(12. \)
- Is \((\mathbb{Z}_{12} ,+ mod (12) )\) cyclic group? If so, what are the possible generators? How many distinct generators exist? What can you say about the number of generators, considering the Euler totient function, and the possible generators in relation to divisors?
- Let \(G=\langle g \rangle \) be a cyclic group with \(|g|=n\). Then \(G=\langle g^k \rangle \) if and only if \(gcd(k,n)=1.\)
- Find all subgroups of \(\mathbb{Z}_{12}\) generated by each element. What are their respective orders? What can you say about these orders?
Consider the symmetric group \(S_4\).
- Find the centre of \(S_4\).
- If possible, find a cyclic subgroup of \(S_4\) with an order of 4.
- Find a non-cyclic subgroup of \(S_4\) with order 4 if possible.
Consider \(\bf{Z}_8\) and the operation \(+ \,mod\,8\).
- Show that \(H=\{0, 2 ,4, 6\}\) is a subgroup of \(\bf{Z}_8\).
- List the left and right cosets of \(H=\{0, 2 ,4, 6\}\) in \(\bf{Z}_8\).
Consider \( D_4=\{e, a, a^2,a^3, b, ab, a^2b, a^3b\}\) with \(a^4=e, b^2=e, ba=a^3b. \)
- Show that \(Z(D_4)=\{e, a^2\}.\) (Hint: \(Z(G)=\{x \in G| ax=xa, \forall a\in G\}.\))
- List the left and right cosets of \(Z(D_4), \) in \(D_4.\)