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.
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$\begingroup$ It is not clear to me what you mean by multiplication. A group has exactly one type of operation, which is sometimes written as $+$ (if the group is abelian) and sometimes written as $\cdot$. Do you mean multiplication in the sense of multiplicative subgroup of a field/ring or more specifically of the real numbers/complex numbers/matrices? $\endgroup$– Jonas LinssenMar 3, 2020 at 7:05
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$\begingroup$ I apologize, I mean under operation · , My professor of my group theory course uses the term "multiplication" often for · $\endgroup$– SlippyMar 3, 2020 at 7:11
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$\begingroup$ No need to apologize :) Multiplication is indeed commonly used to refer to the group operation, the confusion arises when you start to refer to specific examples. For example one could say that the addition of integers is the group multiplication, which sounds odd and is confusing. So my question still is: Do you want any example of a factor group other than $\mathbb Z/n \mathbb Z$ or do you explicitly want groups, whose operation is given by multiplication in some ring? $\endgroup$– Jonas LinssenMar 3, 2020 at 7:33
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2 Answers
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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.
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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)
answered Mar 3, 2020 at 7:44