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Suppose we have $(\mathbb{C},+)$ and $i\mathbb{R}$. Since $(\mathbb{C},+)$ is abelian , we know that $i\mathbb{R}$ is a normal subgroup. Consider quotient group $(\mathbb{C},+)/{i\mathbb{R}}$. Then the set is $\lbrace[a+bi] |a,b \in \mathbb{R} \rbrace=\lbrace(a+bi)+i\mathbb{R} |a,b \in \mathbb{R} \rbrace$. My question is how do we know the quotient set contain what kind of elements ? In this case, how do we know the set contains equvalence classes of complex numbers ?

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Note you can write the coset $(a+bi) + i\Bbb R$ as $ a + i\Bbb R$. We get a different coset for each distinct real number $a$ (the cosets are parallel vertical lines in the plane). –  David Wheeler Jan 31 '13 at 17:23
    
@DavidWheeler: I mean how do we know the element is in $(a+bi)+i\mathbb{R}$ form ? Why it cannot be in other form? –  Idonknow Jan 31 '13 at 17:29
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There are several general notations for elements of a quotient group, and several specialized notations for specific cases. There are many groups that have a reasonable claim to be called a quotient group of $\mathbb{C}$ by $\mathbb{i}\mathbb{R}$.

The particular group you write is a rather standard way of selecting a specific quotient of $\mathbb{C}$ by $\mathbb{i}\mathbb{R}$. It also fits into a rather common set-theoretic pattern for constructing interesting structures:

  • Select a method for aggregating information that lets you uniquely specify what ought to be element of the structure
  • Find a relation that expresses whether two different bits of information ought to specify the same element
  • Define an element of the structure to be an equivalence class of bits of information modulo the aforementioned relation

In this case, the information used to specify an element of the quotient group is by writing an element of $\mathbb{C}$ that is to be its pre-image. And so the elements of (the standard choice of) $\mathbb{C} / i \mathbb{R}$ are the cosets of $i \mathbb{R}$ in $\mathbb{C}$.

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Can I say that elements in quotient group are equivalence classes which defined under some relation on 'big' group ? –  Idonknow Jan 31 '13 at 18:05
    
Yes! the relation $a\sim b$ if and only if $aH = bH$ (or equivalently, if $a^{-1}b \in H$) is an equivalence relation for ANY subgroup $H$ of a group $G$: that is a subgroup PARTITIONS a group into (left) cosets. The requirement that $H$ be normal only enters into it when we wish to impose a group structure on these equivalence classes. Since your "big group" in this case is abelian, it doesn't matter whether you use left or right cosets, and normality is automatic. –  David Wheeler Feb 19 '13 at 22:34
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The answer seems to be: "this is how the quotient group is defined."

If you have doubts whether is definition is meaningful, please present them.

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