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Section 1.10 Quotient groups

By definition, for normal subgroups left and right cosets agree, so we can talk about cosets without specifying left or right. Given a normal subgroup \(H \subset G\text{,}\) we can then define a group structure on the set of cosets. This is called the quotient group:

Consider the collection of all distinct cosets of \(H\) in \(G\text{,}\) which we denote by \(G/H\text{.}\) We want to give it a group structure. We define group multiplication as being the multiplication of sets defined in Definition 1.9.1. Let us first show that the multiplication of two cosets yield another coset. Consider the left cosets \(a H\) and \(b H\) for \(a,b \in G\text{.}\) Elements of \((aH) ( b H)\) have the form \(a h_1 b h_2 = a (h_1 b) h_2\) for \(h_1, h_2 \in H\text{.}\) Since \(H\) is normal, we know that there exists a \(h_3 \in N\) such that \(h_1 b = b h_3\text{.}\) Thus we can write \(a h_1 b h_2 = (a b) h_3 h_2\text{,}\) which is an element of the coset \((a b) H\text{.}\) Hence multiplication of sets defines a group operation \((a H) (b H) = a b H\text{.}\)

Then we can prove that this operation defines a group structure. The identity element is the coset \(e H = H\text{.}\) For any \(g \in G\text{,}\) the inverse of \(g H\) is the coset \(g^{-1} H\text{,}\) since \((g H) (g^{-1} H) = g g^{-1} H = H\text{.}\) Associativity is also clear:

\begin{equation*} ( (a H) (b H) ) (c H) = (a b H) (c H) = a b c H = (a H) (b c H) = (a H) ( (b H) (c H) ). \end{equation*}
Remark 1.10.2.

The reason for this construction to be called a quotient group comes from division of integers. Suppose you consider \(12/4 = 3\text{.}\) You can understand this calculation of the quotient as taking 12 objects, and dividing them into disjoint classes of 4 objects; the result is 3 disjoint classes. Here we are doing the same thing, but instead of only talking about the number of things in a set, we have a group structure, so we start with a group, partition the set into subsets, and equip the resulting partition with the structure of a group.

Note that the quotient group \(G/G\) is clearly isomorphic to the trivial group (since it has only one element, which is the identity); and \(G/\langle e \rangle\) is isomorphic to \(G\text{,}\) since the cosets of the subgroup \(\langle e \rangle\) are just the elements of \(G\) themselves.

Consider the normal subgroup \(A_3 \subset S_3\text{.}\) Using the notation of Definition 1.8.14, there are two cosets, consisting of the even permutations \(E= \{ \pi_1, \pi_5, \pi_6 \}\) and the odd permutations \(O = \{ \pi_2, \pi_3, \pi_4 \}\text{,}\) so the order of the quotient group is 2. We know that there is only one abstract group of order 2, with multiplication table given in Table 1.4.3, and using the group operation on cosets we can check that indeed, \(E \cdot E = O \cdot O = E\text{,}\) \(E \cdot O = O \cdot E = O\text{,}\) since composing two even permutations or two odd permutations gives and even permutation, while composing and odd and an even permutation gives an odd permutation.

A fundamental example of quotient groups consists in the construction of finite cyclic groups as quotients. Start with the integers \(\mathbb{Z}\) under addition. Let \(H = m \mathbb{Z} \) be the subgroup of multiples of the positive integer \(m\text{.}\) Since \(\mathbb{Z}\) is abelian, \(m \mathbb{Z} \) is necessarily normal. We then construct the quotient group \(\mathbb{Z} /m \mathbb{Z}\text{.}\) What do elements of \(\mathbb{Z} / m \mathbb{Z}\) look like? Consider any integer \(k \in \mathbb{Z}\text{.}\) Recalling that the group operation here is addition, the coset \(k \mathbb{Z}\) generated by the integer \(k\) is \(k + m \mathbb{Z}\text{,}\) that is \(k\) plus mutiples of \(m\text{.}\) Since any two \(k_1\) and \(k_2\) that differ by a mutiple of \(m\) generate the same coset, we see that there are exactly \(m\) disjoint cosets \(k + m \mathbb{Z}\text{,}\) generated by \(k \in \{0,1,\ldots,m-1 \}\text{.}\) It follows that the quotient \(\mathbb{Z} / m \mathbb{Z}\) is a finite group of order \(m\text{.}\) Product of cosets (in the sense of Definition 1.9.1)

\begin{equation*} (k_1 + m \mathbb{Z}) \cdot (k_2 + \mathbb{Z}) = k_1 + k_2 + m \mathbb{Z} = k + m \mathbb{Z}, \end{equation*}

where \(k = k_1 + k_2~ (\text{mod } m)\text{.}\)

The group \(\mathbb{Z} / m \mathbb{Z}\) is often denoted by \(\mathbb{Z}_m\text{,}\) and we can write its elements simply as \(\{0,1,\ldots,m-1 \}\text{,}\) each of which representing the corresponding coset, with the understanding that the group operation on \(\mathbb{Z}_m\) is addition modulo \(m\text{.}\) This group if of course finite of order \(m\) and cyclic; it is generated by 1 (the other elements are just the “powers of 1”, with power here meaning repeated application of the operation of addition, i.e. \(2 = 1+1\text{,}\) \(3=1+1+1\text{,}\) etc.). In fact, it can be shown that every cyclic group of order \(m\) is isomorphic to \(\mathbb{Z}_m\text{,}\) so it is of crucial importance. This is also why we can use the same notation \(\mathbb{Z}_m\) as we used for the group of roots of unity; those are the same abstract groups.

Another interesting example is constructed similarly. Start with the real numbers \(\mathbb{R}\text{,}\) and consider the normal subgroup consisting of integers \(\mathbb{Z}\text{.}\) The cosets have the form \(a + \mathbb{Z}\) for \(a \in \mathbb{R}\text{.}\) It is clear that two \(a_1\) and \(a_2\) that are related by the addition of an integer generate the same cosets, hence the cosets are indexed by \(a \in [0,1)\text{.}\) The quotient group \(\mathbb{R} / \mathbb{Z}\) is the group of these cosets, with operation given by adding the cosets \(a_1 + a_2 + \mathbb{Z}\text{,}\) with \(a_1 + a_2\) being understood as addition such that if the result is greater than one, than we substract one so that the result is always between 0 and 1.

Let us now argue that \(\mathbb{R} / \mathbb{Z}\) is isomorphic to the group of complex numbers of absolute value 1 under multiplication, also called \(U(1)\text{.}\) Consider the mapping \(f: \mathbb{R} / \mathbb{Z} \to U(1)\) given by \(f(a + \mathbb{Z}) = \mathrm{e}^{2 \pi i a} \text{.}\) Under this mapping, the group operation on cosets (addition of numbers) is mapped to multiplication of complex numbers. We have not defined group homomorphisms and isomorphisms yet (we will revisit this example in Example 1.11.5), but this mapping is an isomorphism, as it is a bijection, and it preserves the group structure. Thus we can think of \(\mathbb{R} / \mathbb{Z}\) and \(U(1)\) as being the same abstract group. Further, from the point of view of complex numbers of absolute value 1, we can understand \(2 \pi a\) as an angle, and we see that \(U(1)\) is isomorphic to the group of rotations in the complex plane (\(SO(2)\)). The group operation on \(U(1)\text{,}\) multiplication of exponentials, is then mapped to composition of rotations in \(SO(2)\text{.}\) All in all, we obtain that the quotient group \(\mathbb{R} / \mathbb{Z}\) is the same abstract group as the group of two-dimensional rotations \(SO(2)\text{!}\)

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