7.2: The Isomorphism Theorems
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 \(H/N\trianglelefteq G/N\) .