Quotient Groups

This post was machine-translated from the Korean original by Marvin (via Kimi). It may contain errors or awkward phrasing — the Korean original is the source of truth.

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 natural magma structure; moreover, at the end of §Semigroups, Monoids, and Groups we saw that if \(A\) is a group, then the magma \(A/R\) constructed in this way is also a group. This group \(A/R\) is called a quotient group.

Normal Subgroups

On the other hand, from [Set Theory] §Equivalence Relations 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 the set \(G\)

Therefore, we can ask what the condition that \(R\) 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 is exactly \([e]\).

Proposition 1 For a quotient group \(G/R\), the set \([e]\) 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 \(ab^{-1}\sim e\). Thus \(ab^{-1}\in[e]\), so by §Semigroups, Monoids, and Groups, ⁋Proposition 15 we know that \([e]\) is a subgroup.

Conversely, suppose an arbitrary subgroup \(H\) of \(G\) is given. Replacing \([e]\) by \(H\) in the above proof, we can define the following relation.

\[a\sim_{\tiny r}b\iff ab^{-1}\in H\]

It is easy to see that \(\sim_{\tiny r}\) defined in this way is an equivalence relation. In order to define a quotient group via 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 in general \(\sim_{\tiny r}\) need not be left compatible with the operation of \(G\). But if for every \(x\in H\) we have \(cxc^{-1}\in H\) 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 {#rmk} Instead of the equivalence relation \(\sim_r\), if we define 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 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 every \(g\in G\) and every \(h\in H\), we always have \(ghg^{-1}\in H\).

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

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

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

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 \(G/R\) coincide. 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, the relations \(\sim_{\tiny r}\) and \(\sim_{\tiny l}\) obtained above 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 arbitrary \(a\in G\) and its equivalence class \([a]_{\tiny r}\),

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

Thus, 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 as addition, these would conventionally be denoted \(H+a\) 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, given an arbitrary subgroup \(H\) of \(G\), 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 In general \(Ha\neq aH\), but one can easily verify that the necessary and sufficient condition for \(Ha=aH\) to hold for every \(a\in G\) 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 \([G:H]\) of \(H\) 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 \(H\), the identity \(\lvert G\rvert=[G:H]\lvert H\rvert\) holds.

This proposition holds even when \(G\) or \(H\) is infinite, but in the special case when they are finite, we obtain the result that for any subgroup $H$ of a group $G$, the order $\lvert H\rvert$ divides the order $\lvert G\rvert$.

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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. ↩