7.2: The Isomorphism Theorems
- Page ID
- 98009
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The next theorem is arguably the crowning achievement of the course.
Let \(G_1\) and \(G_2\) be groups and suppose \(\phi:G_1\to G_2\) is a homomorphism. Then \[G_1/\ker(\phi)\cong \phi(G_1).\] If \(\phi\) is onto, then \[G_1/\ker(\phi)\cong G_2.\]
Let \(\phi:Q_8\to V_4\) be the homomorphism described in Problem 7.1.6. Use the First Isomorphism Theorem to prove that \(Q_8/\langle-1\rangle\cong V_4\).
For \(n\geq 2\), define \(\phi:S_n\to \mathbb{Z}_2\) via \[\phi(\sigma)=\begin{cases} 0, & \sigma \text{ even}\\ 1, & \sigma \text{ odd}. \end{cases}\] Use the First Isomorphism Theorem to prove that \(S_n/A_n\cong \mathbb{Z}_2\).
Use the First Isomorphism Theorem to prove that \(\mathbb{Z}/6\mathbb{Z}\cong \mathbb{Z}_6\). Attempt to draw a picture of this using Cayley diagrams.
Use the First Isomorphism Theorem to prove that \((\mathbb{Z}_4\times \mathbb{Z}_2)/(\{0\}\times \mathbb{Z}_2)\cong \mathbb{Z}_4\).
The next theorem is a generalization of Theorem [thm:orderImage] and follows from the First Isomorphism Theorem together with Lagrange’s Theorem.
Let \(G_1\) and \(G_2\) be groups and suppose \(\phi:G_1\to G_2\) is a homomorphism. If \(G_1\) is finite, then \(|\phi(G_1)|\) divides \(|G_1|\).
We finish the chapter by listing a few of the remaining isomorphism theorems.
Let \(G\) be a group with \(H\leq G\) and \(N\trianglelefteq G\). Then
- \(HN:=\{hn\mid h\in H, n\in N\}\leq G\);
- \(N\trianglelefteq HN\);
- \(H\cap N\trianglelefteq H\);
- \(\displaystyle H/(H\cap N)\cong HN/N\).
Let \(G\) be a group with \(H,K\trianglelefteq G\) and \(K\leq H\). Then \(H/K\trianglelefteq G/K\) and \[G/H\cong (G/K)/(H/K).\]
The last isomorphism theorem is sometimes called the Lattice Isomorphism Theorem.
Let \(G\) be a group with \(N\trianglelefteq G\). Then there is a bijection from the set of subgroups of \(G\) that contain \(N\) onto the set of subgroups of \(G/N\). In particular, every subgroup \(G\) is of the form \(H/N\) for some subgroup \(H\) of \(G\) containing \(N\) (namely, its preimage in \(G\) under the canonical projection homomorphism from \(G\) to \(G/N\).) This bijection has the following properties: for all \(H,K \leq G\) with \(N\leq H\) and \(N\subseteq K\), we have
- \(H\leq K\) if and only if \(H/N \leq K/N\)
- If \(H\leq K\), then \([K:H]=[K/N:H/N]\)
- \(\langle H,K\rangle/N=\langle H/N,K/N\rangle\)
- \((H\cap K)/N=H/N \cap K/N\)
- \(H\trianglelefteq G\) if and only if \(H/N\trianglelefteq G/N\).