Example of the quotient group of a multiplicative group? Could someone offer me an example of the quotient group of a specific (not arbitrary) group under operation multiplication? I am trying to better conceptualize quotient groups via examples, and I have done so with integers modulo n under addition, but I am struggling to find a specific example of a quotient group of a specific group under multiplication. 
 A: Consider $S=\{z\in\mathbb{C}:|z|=1\}$ which is the unit circle in plane. Elements of $S$ are complex numbers of the form $z=e^{it}$ for some $t\in[0,2\pi)$ and for $z_1=e^{it_1},z_2=e^{it_2}\in S$,
$$z_1z_2=e^{it_1}e^{it_2}=\begin{cases} e^{i(t_1+t_2)}&\text{if }t_1+t_2\leq2\pi\\
e^{i(t_1+t_2-2\pi)}&\text{if }t_1+t_2>2\pi \end{cases}$$
ans $S$ is an abelian group with this multiplication. Now try the quotients like $S/H$ where $$H=(e^{2\pi i/n})$$ is the cyclic subgroup generated by $e^{2\pi i/n}$. Note that $|H|<\infty\iff n$ is an integer.
A: Here are some examples of factor groups:


*

*obviously $\mathbb Z / n \mathbb Z$

*$G=(\mathbb R,+), H=2\pi\mathbb Z$ then $G/H \cong (S, \cdot)$, where $S= \{z \in \mathbb C \mid \Vert z \Vert =1\}$ via the group homomorphism $t \mapsto e^{it}$.

*$G = (GL_n(R),*), H = (SL_n(R),*)$, where $SL_n(R) = \{\text{matrices with det }1\}$, yields $G/H \cong R^{*}$ via the group homomorphism $M \mapsto \det M$

*Factor spaces of vector spaces i.e. $V/W$ (just forget scalar multiplication, you have underlying abelian groups)
