7.2: The Isomorphism Theorems

The next theorem is arguably the crowning achievement of the course.

Theorem \(\PageIndex{1}\): The First Isomorphism Theorem

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 \[G_1/\ker(\phi)\cong G_2.\]

Problem \(\PageIndex{1}\)

Let \(\phi:Q_8\to V_4\) be the homomorphism described in Problem 7.1.6. Use the First Isomorphism Theorem to prove that \(Q_8/\langle-1\rangle\cong V_4\).

Problem \(\PageIndex{2}\)

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 \(S_n/A_n\cong \mathbb{Z}_2\).

Problem \(\PageIndex{3}\)

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.

Problem \(\PageIndex{4}\)

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.

Theorem \(\PageIndex{2}\)

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.

Theorem \(\PageIndex{3}\): The Second Isomorphism Theorem

Let \(G\) be a group with \(H\leq G\) and \(N\trianglelefteq G\). Then

1. \(HN:=\{hn\mid h\in H, n\in N\}\leq G\);
2. \(N\trianglelefteq HN\);
3. \(H\cap N\trianglelefteq H\);
4. \(\displaystyle H/(H\cap N)\cong HN/N\).

Theorem \(\PageIndex{4}\): The Third Isomorphism Theorem

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.

Theorem \(\PageIndex{5}\): The Fourth 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

1. \(H\leq K\) if and only if \(H/N \leq K/N\)
2. If \(H\leq K\), then \([K:H]=[K/N:H/N]\)
3. \(\langle H,K\rangle/N=\langle H/N,K/N\rangle\)
4. \((H\cap K)/N=H/N \cap K/N\)
5. \(H\trianglelefteq G\) if and only if \(H/N\trianglelefteq G/N\).