This post was translated from Korean by LLM (Kimi). The translation may contain errors or awkward sentences. The Korean original is the source of truth.
Previously, in §Algebraic Structures, §§Quotient Structures, we proved that when an equivalence relation \(R\) is compatible with the operation of a magma \(A\), the quotient set \(A/R\) can be given a magma structure in a natural way. Moreover, at the end of §Semigroups, Monoids, Groups, we saw that when \(A\) is a group, the magma \(A/R\) constructed in this way is also a group. This group \(A/R\) is called a quotient group.
Normal Subgroups
Meanwhile, from [Set Theory] §Equivalence Relations, ⁋Proposition 7, we know that the following two are equivalent:
Giving an equivalence relation \(R\) on a set \(G\) \(\iff\) Choosing a partition \((G_i)_{i\in I}\) of \(G\)
Therefore, we can consider what requiring the equivalence relation \(R\) to be compatible with the operation of \(G\) means on the right-hand side.
First, assume that \(R\) is compatible with the operation of \(G\). Then each element of \(G/R\) forms a partition of \(G\), and in particular, the set containing the identity element is uniquely \([e]\).
Proposition 1 For the quotient group \(G/R\), \([e]\) is a subgroup of \(G\).
Proof
Let \(a,b\in [e]\). That is, \(a\sim e\sim b\). Now since \(R\) is compatible with the operation of \(G\), multiplying both sides of \(a\sim b\) on the right by \(b^{-1}\) yields \(ab^{-1}\sim e\). Thus \(ab^{-1}\in[e]\), and by §Semigroups, Monoids, Groups, ⁋Proposition 15, we know that \([e]\) is a subgroup.
Conversely, suppose an arbitrary subgroup \(H\) of \(G\) is given. Replacing \([e]\) with \(H\) in the above proof, we can define the following relation:
\[a\sim_{\tiny r}b\iff ab^{-1}\in H\]It is easy to show that \(\sim_{\tiny r}\) defined in this way is an equivalence relation. To define a quotient group through this, this equivalence relation must be compatible with the operation of \(G\). Let arbitrary \(a,b,c\in G\) be given. First, if \(a\sim_{\tiny r}b\) holds, then
\[(ac)(bc)^{-1}=acc^{-1}b^{-1}=ab^{-1}\in H\]so \(ac\sim_{\tiny r} bc\) holds. That is, \(\sim_{\tiny r}\) is right compatible with the operation of \(G\). However,
\[(ca)(cb)^{-1}=cab^{-1}c^{-1}\]so \(\sim_{\tiny r}\) is not necessarily left compatible with the operation of \(G\) in general. But if for arbitrary \(x\in H\), \(cxc^{-1}\in H\) holds for all \(c\in G\), then the right-hand side becomes an element of \(H\), and thus \(\sim_{\tiny r}\) defines a compatible equivalence relation on \(G\).
Remark If instead of the equivalence relation \(\sim_r\) we define the following relation
\[a\sim_{\tiny l} b\iff a^{-1}b\in H\]then \(\sim_{\tiny l}\) is left compatible, and since
\[(ac)^{-1}(bc)=c^{-1}(a^{-1}b)c\]it is not right compatible. For this relation to be right compatible, \(c^{-1}xc\in H\) must hold for arbitrary \(c\in G\) and arbitrary \(x\in H\), which is the same condition obtained above.
Definition 2 A subgroup \(H\) of a group \(G\) is called a normal subgroup if for arbitrary \(g\in G\) and arbitrary \(h\in H\), \(ghg^{-1}\in H\) always holds.
Meanwhile, since \(g\) can be chosen arbitrarily, we can show that \(H\) being a normal subgroup is equivalent to \(gHg^{-1}=H\) holding for arbitrary \(g\). By the above discussion, when a normal subgroup \(H\) of \(G\) is given, we can obtain the corresponding quotient group. The resulting quotient group is denoted by \(G/H\).
From Proposition 1, for arbitrary \(a\in [e]\), the following
\[a\sim e\implies gag^{-1}\sim geg^{-1}=e\]shows that \([e]\) is a normal subgroup. Also, when we set \(H=[e]\), the corresponding \(\sim_{\tiny r}\) is exactly the same as the original equivalence relation \(\sim\), so \(G/H\) and \(G/R\) are equal. Conversely, for \(\sim_{\tiny r}\) defined from an arbitrary normal subgroup \(H\), \(G/H=G/{\sim_{\tiny r}}\) also holds. From this, we know that giving a compatible equivalence relation on \(G\) is the same as choosing a normal subgroup of \(G\).
Cosets
Now consider a group \(G\) and an arbitrary subgroup \(H\). Even if \(H\) is not normal, \(\sim_{\tiny r}\) and \(\sim_{\tiny l}\) obtained from the above discussion are still equivalence relations, so we can examine what the quotient sets \(G/{\sim_{\tiny r}}\) and \(G/{\sim_{\tiny l}}\) look like.
First, consider the elements of \(G/{\sim_{\tiny r}}\). For arbitrary \(a\in G\) and its equivalence class \([a]_{\tiny r}\), we know that
\[x\in [a]_{\tiny r}\iff x\sim_{\tiny r} a\iff xa^{-1}\in H\]Therefore, if we define the set \(Ha\) by the following formula
\[Ha:=\{ha\mid h\in H\}\]then \([a]_{\tiny r}=Ha\) holds. Similarly, for \(G/{\sim_{\tiny l}}\), \([a]_{\tiny l}=aH\) holds. Of course, if the operation of \(G\) were written as addition, these would conventionally be denoted as \(H+a\) and \(a+H\) respectively.
Definition 3 The two sets \(Ha\) and \(aH\) defined above are called a right coset and a left coset, respectively.
Thus, when an arbitrary subgroup \(H\) of \(G\) is given, the two equivalence relations \(\sim_{\tiny r}\) and \(\sim_{\tiny l}\) partition \(G\) into right cosets and left cosets, respectively. In this case, the quotient set of \(G\) by \(\sim_{\tiny r}\) is denoted \(H\setminus G\), and the quotient set of \(G\) by \(\sim_{\tiny l}\) is denoted \(G/H\).1 Although \(Ha\neq aH\) in general, it is easy to verify that \(Ha=aH\) holds if and only if \(H\) is normal.
Moreover, for arbitrary \(a\in G\),
\[{a\cdot}: H\rightarrow aH;\quad h\mapsto ah,\qquad {a^{-1}\cdot}: aH\rightarrow H;\quad ah\mapsto h\]are inverses of each other, so we can see that both right cosets and left cosets have the same cardinality as \(H\). Also, defining a function \(H\setminus G\rightarrow G/H\) by the formula
\[Ha\mapsto a^{-1}H\]it is easy to verify that this function is bijective. That is, \(\lvert H\setminus G\rvert=\lvert G/H\rvert\).
Definition 4 For a group \(G\) and a subgroup \(H\), the index \([G:H]\) of \(H\) is defined as \(\lvert G/H\rvert\).
From the structure of \(G/H\) examined above and the size of each element of \(G/H\), the following proposition is obvious.
Proposition 5 (Lagrange) For a group \(G\) and a subgroup \(H\), \(\lvert G\rvert=[G:H]\lvert H\rvert\) holds.
This proposition holds even when \(G\) or \(H\) is infinite, but particularly when they are finite, we obtain the result that for any subgroup \(H\) of a group \(G\), \(\lvert H\rvert\) is a divisor of \(\lvert G\rvert\).
References
[Bou] Bourbaki, N. Algebra I. Elements of Mathematics. Springer. 1998.
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The notation for right cosets overlaps with that for set difference, but since there will not be many occasions to use right cosets, we will not introduce a separate notation. ↩
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