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.

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, and Groups, we saw that if \(A\) is a group, then 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 §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 \(G\)

Hence we may ask what the requirement that the equivalence relation \(R\) 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 the elements of \(G/R\) form a partition of \(G\), and in particular the part containing the identity element 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 \(b^{-1}\) yields \(ab^{-1}\sim e\). 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 \([e]\) with \(H\) in the argument above, we can define the following relation:

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

It is easy to verify that \(\sim_{\tiny r}\) defined in this way is an equivalence relation. To define a quotient group through this relation, we need this equivalence relation to be compatible with the operation of \(G\). Let arbitrary \(a,b,c\in G\) be given. First, if \(a\sim_{\tiny r}b\) holds,

\[(ac)(bc)^{-1}=acc^{-1}b^{-1}=ab^{-1}\in H\]

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 \(x\in H\), \(cxc^{-1}\in H\) holds for all \(c\in G\), then the right-hand side belongs to \(H\), 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 \(\sim_{\tiny l}\) would be 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 \(h\in H\), \(ghg^{-1}\in H\) always holds.

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, when a normal subgroup \(H\) of \(G\) is given, we can obtain the corresponding quotient group. The resulting quotient group is then written as \(G/H\).

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

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

shows that \([e]\) is a normal subgroup. Also, when \(H=[e]\), the corresponding \(\sim_{\tiny r}\) is exactly 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 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, consider the elements of \(G/{\sim_{\tiny r}}\). For arbitrary \(a\in G\) and its equivalence class \([a]_{\tiny r}\), we know that

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

Thus, 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 \(G\) were written as addition, these would conventionally be written 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 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 written as \(H\setminus G\), and the quotient set of \(G\) by \(\sim_{\tiny l}\) is written as \(G/H\).¹ In general \(Ha\neq aH\), but it is easy to check 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 mutually inverse, so all right cosets and left cosets 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 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 \(H\), we define the index \([G:H]\) of \(H\) 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 also holds when \(G\) or \(H\) is infinite, but in particular when they are finite, we obtain the result that for any subgroup $$H$$ of a group $$G$$, $$\lvert H\rvert$$ divides $$\lvert G\rvert$$.

---

References

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

---

1. The notation for right cosets conflicts with set difference notation, but since there will not be much occasion to use right cosets, we shall not introduce a separate notation. ↩