Homomorphisms and Quotient Groups - Applications

The Second and Third Isomorphism Theorems provide additional structural insights about how subgroups, quotients, and homomorphisms interact.

TheoremSecond Isomorphism Theorem

Let $H \leq G$ be a subgroup and $N \trianglelefteq G$ a normal subgroup. Then:

$HN = \{hn : h \in H, n \in N\}$ is a subgroup of $G$
$N \trianglelefteq HN$ and $H \cap N \trianglelefteq H$
$H/(H \cap N) \cong HN/N$

This theorem is sometimes called the Diamond Isomorphism Theorem because the subgroups form a diamond-shaped lattice diagram. The isomorphism identifies $H/(H \cap N)$ with the "shadow" of $H$ in the quotient $G/N$ .

ExampleApplication to Matrix Groups

Let $H = SO_n(\mathbb{R})$ (special orthogonal matrices) and $N = \{\pm I\}$ (scalar matrices $\pm 1$ ) inside $O_n(\mathbb{R})$ . Then:

$HN = O_n(\mathbb{R})$ (all orthogonal matrices)
$H \cap N = \{\pm I\} \cap SO_n(\mathbb{R})$

For $n$ even, $H \cap N = \{I\}$ , giving $SO_n(\mathbb{R}) \cong O_n(\mathbb{R})/\{\pm I\}$ .

TheoremThird Isomorphism Theorem

Let $K, N \trianglelefteq G$ with $K \subseteq N$ . Then:

$K \trianglelefteq N$ and $N/K \trianglelefteq G/K$
$(G/K)/(N/K) \cong G/N$

This theorem allows us to "compose" quotients. Taking quotients in stages $G \to G/K \to G/N$ is the same as taking the quotient $G \to G/N$ directly.

TheoremCorrespondence Theorem

Let $N \trianglelefteq G$ . The map $\Phi: \{H : N \subseteq H \leq G\} \to \{K : K \leq G/N\}$ given by $\Phi(H) = H/N$ is a bijection that preserves:

Inclusion: $H_1 \subseteq H_2 \iff H_1/N \subseteq H_2/N$
Normality: $H \trianglelefteq G \iff H/N \trianglelefteq G/N$
Index: $[G:H] = [G/N:H/N]$

Remark

These theorems show that the subgroup lattice of $G/N$ is precisely the upper part of the lattice of $G$ (above $N$ ). This makes it possible to analyze quotient groups by understanding the original group's subgroup structure.

Together, the isomorphism theorems form a complete toolkit for manipulating quotients and homomorphisms. They appear constantly in proofs, often implicitly, whenever we need to relate different group constructions.