Some Examples of Quotient Groups

# Some Examples of Quotient Groups

Recall from the Quotient Groups page that if $(G, *)$ is a group and $(H, *)$ is a normal subgroup then we let $G / H = \{ gH : g \in G \}$ (i.e., $G / H$ is simply the set of left cosets of $H$ with representatives $g \in G$), then the Quotient of $G$ by $H$ is the group $(G / H, \cdot)$ where $\cdot$ is the binary operation on $G / H$ defined for all $g_1 H, g_2 H \in G / H$ by:

(1)
\begin{align} \quad (g_1 H) \cdot (g_2 H) = (g_1 * g_2)H \end{align}

We saw that $(G / H, \cdot)$ is indeed a group with this operation. We will now look at some simple examples of quotient groups.

First, let $(G, *)$ be any group. If $e$ is the identity on $G$ then $( \{ e \}, *)$ is the trivial subgroup of $(G, *)$ and is also a normal subgroup of $(G, *)$. So we can take the quotient of $G$ by $\{ e \}$. We have that $G / \{ e \} = \{ g \{ e \} : g \in G \} = \{ \{ g \} : g \in G \}$, and for all $g_1, g_2 \in G$:

(2)
\begin{align} \quad (g_1 \{ e \}) \cdot (g_2 \{ e \} ) = (g_1 * g_2) \{ e \} \quad \Leftrightarrow \{ g_1 \} \cdot \{ g_2 \} = \{ g_1 * g_2 \} \end{align}

It is not hard to see that $G / \{ e \}$ is isomorphic to $G$ itself!

For another example, consider the subgroup $(G, *)$, i.e., the whole subgroup. Then $G / G = \{ gG : g \in G \} = \{ G \}$. Since $G / G$ contains only one element is must be isomorphic to the trivial subgroup $( \{ e \}, *)$.

For a third example, consider the group $(\mathbb{Z}, +)$ and the subgroup $(n\mathbb{Z}, +)$ where $n \in \mathbb{N}$. Notice that $(\mathbb{Z}, +)$ is an abelian group since for all integers $a, b \in \mathbb{Z}$, $a + b = b + a$. So we know that every subgroup of $(\mathbb{Z}, +)$ is normal. In particular, the subgroup $(n\mathbb{Z}, +)$ (where $n\mathbb{Z} = \{ ..., -2n, -n, 0, n, 2n, ... \}$) is normal. For all $(a + n\mathbb{Z}), (b + n\mathbb{Z}) \in \mathbb{Z} / n\mathbb{Z}$ we have that:

(3)
\begin{align} \quad (a + n\mathbb{Z}) \cdot (b + n\mathbb{Z}) = (a + b) + n \mathbb{Z} \end{align}

So $\mathbb{Z} / n\mathbb{Z}$ is isomorphic to $\mathbb{Z}_n$ with the operation $+$.