Assignment 6
- Page ID
- 169956
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- Find subgroups \( H \) and \( K \) of \( D_4 \) such that \( K \) is normal to \( H \) and \(H\) is normal to \(D_4\) but \( K \) is not normal to \(D_4\).
- Find subgroups \( H \) and \( K \) of \( S_4 \) such that \( K \) is normal to \( H \) and \(H\) is normal to \(S_4\) but \( K \) is not normal to \(S_4.\)
- Prove or disprove: If all the subgroups of a group \(G\) are normal subgroups of \(G\), then \(G\) is abelian.
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Let \(G=<a>\) where \(|a|=12,\) and let \(H=<a^4>.\) Find all cosets in \(G/H\) and write down the Cayley table. Is \(G/H\) cyclic? Why or why not?
Find the cosets in \(D_6/Z(D_6),\) write down the Cayley table of \(D_6/Z(D_6).\) Is \(D_6/Z(D_6)\) cyclic, why or why not? -
Prove or disprove the following statements: If \(H \) is a normal subgroup of \(G\) such that \(H\) and \(G/H\) are abelian, then
\(G\) is abelian.
- Let \(G\) be a group. Show that the center of the group \(Z(G)\) is a normal subgroup of \(G.\)
- If \(G/Z(G)\) is cyclic , then \(G\) is abelian.
- Let \(G\) be a non-ablian group of order \(pq\), where \(p\) and \(q\) are prime. Then the center of \(G\), \(Z(G)=\{e\}.\)
- Let \(h: G\to G_1\) be a group homomorphism. If \(K\) is normal to subgroup of \(G,\) then show that \(h(K)\) is a normal subgroup of\(h(G).\)
- If \(h_1: G\to G_1\) is a group homomorphism and \(G=\langle X\rangle\) is generated by a subset \(X\), then show that \(h=h_1\) if and only if \(h(x)=h_1(x), \forall x\in X.\)
- Show that there are almost six homomorphisms from \(S_3\) to a cyclic group of order \(6\), \(\mathbb{Z}_6\).
Let \(G\) be a group and let \(a \in G\) then show that \(\sigma_a :G \to G\) defined by \(\sigma_a(g)=aga^{-1}\)is an isormorphism.
Prove or disprove the following statements:
- \(U(5)\) is isormorphic to \( \mathbb{Z}_4\) .
- \(\mathbb{Z}\) under addition is isormorphic to \(\mathbb{Q}\) under addition.
List or characterize all of the following rings' units. Justify your answer.
- \(\mathbb{Z}_{12}\)
- \(M_{22}(\mathbb{Z}_2)\)
- \(\mathbb{Z}(i)\)
- Answer
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TBA
An element \(e\) of a ring \(R\) is said to be idempotent if \(e^2=e.\)
- If \(e\) is an idempotent of a ring \(R\), then show that \(1-2e\) is a unit.
- Find all idempotents in \(\mathbb{Z}_{12}\).
- Find all the idempotents in an integral domain \(R\).
- Answer
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TBA