Quotient Groups

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 naturally equipped with a magma structure. Moreover, at the end of §Semigroups, Monoids, Groups, we saw that if \(A\) is a group, the magma \(A/R\) constructed in this way is also a group. In this case, the group \(A/R\) is called a quotient group.

Normal Subgroups

On the other hand, from [Set Theory] §Equivalence Relations, ⁋Proposition 7, we know that the following two notions are equivalent.

Giving an equivalence relation \(R\) on a set \(G\) \(\iff\) Choosing a partition \((G_i)_{i\in I}\) of the set \(G\)

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 the elements of \(G/R\) form a partition of \(G\), and in particular, the set containing the identity element is exactly \([e]\).

Proposition 1 For a quotient group \(G/R\), \([e]\) is a subgroup of \(G\).

Proof

Let \(a,b\in [e]\). That is, \(a\sim e\sim b\). 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, \([e]\) 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 using 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, since

\[(ca)(cb)^{-1}=cab^{-1}c^{-1}\]

in general, \(\sim_{\tiny r}\) need not be left compatible with the operation of \(G\). But if for any \(x\in H\), \(cxc^{-1}\in H\) 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 had defined the 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 all \(c\in G\) and all \(x\in H\), which is the same condition obtained above.

Definition 2 A subgroup \(H\) of a group \(G\) is a normal subgroup if for any \(g\in G\) and any \(h\in H\), \(ghg^{-1}\in H\) always holds.

On the other hand, since \(g\) can be chosen arbitrarily, we can show that \(H\) being a normal subgroup is equivalent to \(gHg^{-1}=H\) holding for any \(g\). From the above discussion, when a normal subgroup \(H\) of \(G\) is given, we can obtain the corresponding quotient group. The quotient group obtained in this case is denoted by \(G/H\).

In Proposition 1, from the equation

\[a\sim e\implies gag^{-1}\sim geg^{-1}=e\]

for any \(a\in [e]\), we see that \([e]\) is a normal subgroup. Also, when setting \(H=[e]\), the corresponding \(\sim_{\tiny r}\) is exactly the original equivalence relation \(\sim\), so \(G/H\) and \(G/R\) are equal. Conversely, for any normal subgroup \(H\), \(G/H=G/{\sim_{\tiny r}}\) also holds. From this, we see 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, the \(\sim_{\tiny r}\) 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, let us consider the elements of \(G/{\sim_{\tiny r}}\). For any \(a\in G\) and its equivalence class \([a]_{\tiny r}\),

\[x\in [a]_{\tiny r}\iff x\sim_{\tiny r} a\iff xa^{-1}\in H\]

is observed. Therefore, defining the set \(Ha\) by

\[Ha:=\{ha\mid h\in H\}\]

we have \([a]_{\tiny r}=Ha\). Similarly, for \(G/{\sim_{\tiny l}}\), we have \([a]_{\tiny l}=aH\). Of course, if the operation of \(G\) is written as addition, it is customary to write these as \(H+a\) and \(a+H\), respectively.

Definition 3 The two sets \(Ha\) and \(aH\) defined above are called the right coset and left coset, respectively.

Thus, when any 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 by \(H\setminus G\), and the quotient set of \(G\) by \(\sim_{\tiny l}\) is denoted by \(G/H\).¹ In general, \(Ha\neq aH\), but it is easy to verify that the necessary and sufficient condition for \(Ha=aH\) is that \(H\) is normal.

Moreover, for any \(a\in G\),

\[{a\cdot}: H\rightarrow aH;\quad h\mapsto ah,\qquad {a^{-1}\cdot}: aH\rightarrow H;\quad ah\mapsto h\]

are inverses of each other, so we see that right cosets and left cosets all have the same cardinality as \(H\). Also, defining a function \(H\setminus G\rightarrow G/H\) by

\[Ha\mapsto a^{-1}H\]

it is easy to verify that this function is a bijection. 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 immediate.

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 finite group \(G\), \(\lvert H\rvert\) divides \(\lvert G\rvert\).

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References

[Bou] Bourbaki, N. Algebra I. Elements of Mathematics. Springer. 1998.

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1. The notation for right cosets coincides with the notation for set difference, but since we will not use right cosets often, we will not introduce a separate notation. ↩