10.4: Exercises
- Page ID
- 81108
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)For each of the following groups \(G\text{,}\) determine whether \(H\) is a normal subgroup of \(G\text{.}\) If \(H\) is a normal subgroup, write out a Cayley table for the factor group \(G/H\text{.}\)
- \(G = S_4\) and \(H = A_4\)
- \(G = A_5\) and \(H = \{ (1), (1 \, 2 \, 3), (1 \, 3 \, 2) \}\)
- \(G = S_4\) and \(H = D_4\)
- \(G = Q_8\) and \(H = \{ 1, -1, I, -I \}\)
- \(G = {\mathbb Z}\) and \(H = 5 {\mathbb Z}\)
Find all the subgroups of \(D_4\text{.}\) Which subgroups are normal? What are all the factor groups of \(D_4\) up to isomorphism?
Find all the subgroups of the quaternion group, \(Q_8\text{.}\) Which subgroups are normal? What are all the factor groups of \(Q_8\) up to isomorphism?
Let \(T\) be the group of nonsingular upper triangular \(2 \times 2\) matrices with entries in \({\mathbb R}\text{;}\) that is, matrices of the form
\[ \begin{pmatrix} a & b \\ 0 & c \end{pmatrix}\text{,} \nonumber \]
where \(a\text{,}\) \(b\text{,}\) \(c \in {\mathbb R}\) and \(ac \neq 0\text{.}\) Let \(U\) consist of matrices of the form
\[ \begin{pmatrix} 1 & x \\ 0 & 1 \end{pmatrix}\text{,} \nonumber \]
where \(x \in {\mathbb R}\text{.}\)
- Show that \(U\) is a subgroup of \(T\text{.}\)
- Prove that \(U\) is abelian.
- Prove that \(U\) is normal in \(T\text{.}\)
- Show that \(T/U\) is abelian.
- Is \(T\) normal in \(GL_2( {\mathbb R})\text{?}\)
Show that the intersection of two normal subgroups is a normal subgroup.
If \(G\) is abelian, prove that \(G/H\) must also be abelian.
Prove or disprove: If \(H\) is a normal subgroup of \(G\) such that \(H\) and \(G/H\) are abelian, then \(G\) is abelian.
If \(G\) is cyclic, prove that \(G/H\) must also be cyclic.
Prove or disprove: If \(H\) and \(G/H\) are cyclic, then \(G\) is cyclic.
Let \(H\) be a subgroup of index \(2\) of a group \(G\text{.}\) Prove that \(H\) must be a normal subgroup of \(G\text{.}\) Conclude that \(S_n\) is not simple for \(n \geq 3\text{.}\)
If a group \(G\) has exactly one subgroup \(H\) of order \(k\text{,}\) prove that \(H\) is normal in \(G\text{.}\)
Define the centralizer of an element \(g\) in a group \(G\) to be the set
\[ C(g) = \{ x \in G : xg = gx \}\text{.} \nonumber \]
Show that \(C(g)\) is a subgroup of \(G\text{.}\) If \(g\) generates a normal subgroup of \(G\text{,}\) prove that \(C(g)\) is normal in \(G\text{.}\)
Recall that the center of a group \(G\) is the set
\[ Z(G) = \{ x \in G : xg = gx \text{ for all } g \in G \}\text{.} \nonumber \]
- Calculate the center of \(S_3\text{.}\)
- Calculate the center of \(GL_2 ( {\mathbb R} )\text{.}\)
- Show that the center of any group \(G\) is a normal subgroup of \(G\text{.}\)
- If \(G / Z(G)\) is cyclic, show that \(G\) is abelian.
Let \(G\) be a group and let \(G' = \langle aba^{- 1} b^{-1} \rangle\text{;}\) that is, \(G'\) is the subgroup of all finite products of elements in \(G\) of the form \(aba^{-1}b^{-1}\text{.}\) The subgroup \(G'\) is called the commutator subgroup of \(G\text{.}\)
- Show that \(G'\) is a normal subgroup of \(G\text{.}\)
- Let \(N\) be a normal subgroup of \(G\text{.}\) Prove that \(G/N\) is abelian if and only if \(N\) contains the commutator subgroup of \(G\text{.}\)