Quotient Groups

We previously proved in §Algebraic Structures, §§Quotient Structures that when an equivalence relation \(R\) is compatible with the operation of a magma \(A\), the quotient set \(A/R\) can be endowed with a magma structure in a natural way; moreover, at the end of §Semigroups, Monoids, and Groups, we saw that when \(A\) is a group, the magma \(A/R\) constructed in this way also becomes a group. In this case, the group \(A/R\) is called a quotient group.

Normal Subgroups

Meanwhile, through [Set Theory] §Equivalence Relations, we know that the following two are equivalent:

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

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 only set among them containing the identity is \([e]\).

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 \(b^{-1}\) gives \(ab^{-1}\sim e\). That is, \(ab^{-1}\in[e]\), so by §Semigroups, Monoids, and Groups, ⁋Proposition 15 we know that \([e]\) is a subgroup.

Conversely, for \(G\), suppose an arbitrary subgroup \(H\) is given. Replacing \([e]\) in the above proof with \(H\), 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. In order to define a quotient group through this, this equivalence relation must be compatible with the operation of \(G\). Suppose arbitrary \(a,b,c\in G\) are 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 every \(x\in H\), \(cxc^{-1}\in H\) holds for all \(c\in G\), then the right-hand side will be an element of \(H\), and thus \(\sim_{\tiny r}\) defines a compatible equivalence relation on \(G\).

::: Remark {#rmk} 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. In order for this relation to be right compatible, for every \(c\in G\) and every \(x\in H\), \(c^{-1}xc\in H\) must hold, which is the same as the condition obtained above. :::

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

Meanwhile, since \(g\) can be chosen arbitrarily, one can show that \(H\) being a normal subgroup is equivalent to having, for every \(g\), \(gHg^{-1}=H\). By the above discussion, for \(G\), given a normal subgroup \(H\), we obtain the corresponding quotient group. This quotient group is denoted \(G/H\).

From Proposition 1, for any \(a\in [e]\), from the formula

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

we know that \([e]\) is a normal subgroup. Also, when setting \(H=[e]\), the corresponding \(\sim_{\tiny r}\) is exactly identical to the original equivalence relation \(\sim\), so \(G/H\) and \(G/R\) are equal to each other. Conversely, for any normal subgroup \(H\), the relation \(\sim_{\tiny r}\) defined from it also satisfies \(G/H=G/{\sim_{\tiny r}}\). From this, we know that giving a compatible equivalence relation on \(G\) is equivalent to 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 in the discussion above are nonetheless 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 \([a]_{\tiny r}\), we know that

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

holds. Therefore, defining the set \(Ha\) by the formula

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

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

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

Therefore, for \(G\), given an arbitrary subgroup \(H\), the two equivalence relations \(\sim_{\tiny r}\) and \(\sim_{\tiny l}\) partition \(G\) into right cosets and left cosets, respectively. In this case, with respect to \(\sim_{\tiny r}\), the quotient set of \(G\) is denoted \(H\setminus G\), and with respect to \(\sim_{\tiny l}\), the quotient set of \(G\) is denoted \(G/H\).¹ In general, \(Ha\neq aH\), but one can easily verify that for any \(a\in G\), the necessary and sufficient condition for \(Ha=aH\) to hold is that \(H\) is normal.

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

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

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

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

one can easily check 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 of \(H\), denoted \([G:H]\), is defined to be \(\lvert G/H\rvert\).

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 an infinite set, but especially when they are finite, we obtain the result that for a group \(G\) and any subgroup \(H\), \(\lvert H\rvert\) is a divisor of \(\lvert G\rvert\).

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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 conflicts with that for set difference, but since right cosets will not be used much, we shall not introduce a separate notation. ↩