Quotient Groups

Quotient Groups

Normal Subgroups

A subgroup $N$ of $G$ is normal (written $N \trianglelefteq G$) if $aNa^{-1} = N$ for all $a \in G$. In an abelian group, every subgroup is normal.

Since $\mathbb{Z}_n$ is abelian, all subgroups of $\mathbb{Z}_n$ are normal.

The Quotient Group

If $N \trianglelefteq G$, the quotient group $G/N$ is the set of cosets of $N$ in $G$ with the operation:

$(aN)(bN) = (ab)N$

This is well-defined precisely because $N$ is normal.

Example: $\mathbb{Z}_6 / \{0, 3\}$

The cosets are $\{0, 3\}$, $\{1, 4\}$, $\{2, 5\}$. Labeling them $\bar{0}, \bar{1}, \bar{2}$:

This is isomorphic to $\mathbb{Z}_3$!

First Isomorphism Theorem

If $\phi: G \to H$ is a homomorphism, then:

$G / \ker(\phi) \cong \text{im}(\phi)$

This is one of the most important theorems in algebra.

Your Task

Implement quotient_cayley_table(n, subgroup) that computes the Cayley table for $\mathbb{Z}_n / H$ where $H$ is a subgroup. Represent each coset by its smallest element. Return the table as a list of lists.