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Section9.2Direct Products

Given two groups \(G\) and \(H\), it is possible to construct a new group from the Cartesian product of \(G\) and \(H\), \(G \times H\). Conversely, given a large group, it is sometimes possible to decompose the group; that is, a group is sometimes isomorphic to the direct product of two smaller groups. Rather than studying a large group \(G\), it is often easier to study the component groups of \(G\).

SubsectionExternal Direct Products

If \((G,\cdot)\) and \((H, \circ)\) are groups, then we can make the Cartesian product of \(G\) and \(H\) into a new group. As a set, our group is just the ordered pairs \((g, h) \in G \times H\) where \(g \in G\) and \(h \in H\). We can define a binary operation on \(G \times H\) by \begin{equation*}(g_1, h_1)(g_2, h_2) = (g_1 \cdot g_2, h_1 \circ h_2);\end{equation*} that is, we just multiply elements in the first coordinate as we do in \(G\) and elements in the second coordinate as we do in \(H\). We have specified the particular operations \(\cdot\) and \(\circ\) in each group here for the sake of clarity; we usually just write \((g_1, h_1)(g_2, h_2) = (g_1 g_2, h_1 h_2)\).

Example9.14

Let \({\mathbb R}\) be the group of real numbers under addition. The Cartesian product of \({\mathbb R}\) with itself, \({\mathbb R} \times {\mathbb R} = {\mathbb R}^2\), is also a group, in which the group operation is just addition in each coordinate; that is, \((a, b) + (c, d) = (a + c, b + d)\). The identity is \((0,0)\) and the inverse of \((a, b)\) is \((-a, -b)\).

Example9.15

Consider \begin{equation*}{\mathbb Z}_2 \times {\mathbb Z}_2 = \{ (0, 0), (0, 1), (1, 0),(1, 1) \}.\end{equation*} Although \({\mathbb Z}_2 \times {\mathbb Z}_2\) and \({\mathbb Z}_4\) both contain four elements, they are not isomorphic. Every element \((a,b)\) in \({\mathbb Z}_2 \times {\mathbb Z}_2\) has order 2, since \((a,b) + (a,b) = (0,0)\); however, \({\mathbb Z}_4\) is cyclic.

The group \(G \times H\) is called the external direct product of \(G\) and \(H\). Notice that there is nothing special about the fact that we have used only two groups to build a new group. The direct product \begin{equation*}\prod_{i = 1}^n G_i = G_1 \times G_2 \times \cdots \times G_n\end{equation*} of the groups \(G_1, G_2, \ldots, G_n\) is defined in exactly the same manner. If \(G = G_1 = G_2 = \cdots = G_n\), we often write \(G^n\) instead of \(G_1 \times G_2 \times \cdots \times G_n\).

Example9.16

The group \({\mathbb Z}_2^n\), considered as a set, is just the set of all binary \(n\)-tuples. The group operation is the “exclusive or” of two binary \(n\)-tuples. For example, \begin{equation*}(01011101) + (01001011) = (00010110).\end{equation*} This group is important in coding theory, in cryptography, and in many areas of computer science.

Proof
Example9.19

Let \((8, 56) \in {\mathbb Z}_{12} \times {\mathbb Z}_{60}\). Since \(\gcd(8,12) = 4\), the order of 8 is \(12/4 = 3\) in \({\mathbb Z}_{12}\). Similarly, the order of \(56\) in \({\mathbb Z}_{60}\) is \(15\). The least common multiple of 3 and 15 is 15; hence, \((8, 56)\) has order 15 in \({\mathbb Z}_{12} \times {\mathbb Z}_{60}\).

Example9.20

The group \({\mathbb Z}_2 \times {\mathbb Z}_3\) consists of the pairs \begin{align*} & (0,0), & & (0, 1), & & (0, 2), & & (1,0), & & (1, 1), & & (1, 2). \end{align*} In this case, unlike that of \({\mathbb Z}_2 \times {\mathbb Z}_2\) and \({\mathbb Z}_4\), it is true that \({\mathbb Z}_2 \times {\mathbb Z}_3 \cong {\mathbb Z}_6\). We need only show that \({\mathbb Z}_2 \times {\mathbb Z}_3\) is cyclic. It is easy to see that \((1,1)\) is a generator for \({\mathbb Z}_2 \times {\mathbb Z}_3\).

The next theorem tells us exactly when the direct product of two cyclic groups is cyclic.

Proof

In Chapter 13, we will prove that all finite abelian groups are isomorphic to direct products of the form \begin{equation*}{\mathbb Z}_{p_1^{e_1}} \times \cdots \times {\mathbb Z}_{p_k^{e_k}}\end{equation*} where \(p_1, \ldots, p_k\) are (not necessarily distinct) primes.

SubsectionInternal Direct Products

The external direct product of two groups builds a large group out of two smaller groups. We would like to be able to reverse this process and conveniently break down a group into its direct product components; that is, we would like to be able to say when a group is isomorphic to the direct product of two of its subgroups.

Let \(G\) be a group with subgroups \(H\) and \(K\) satisfying the following conditions.

  • \(G = HK = \{ hk : h \in H, k \in K \}\);

  • \(H \cap K = \{ e \}\);

  • \(hk = kh\) for all \(k \in K\) and \(h \in H\).

Then \(G\) is the internal direct product of \(H\) and \(K\).

Example9.24

The group \(U(8)\) is the internal direct product of \begin{equation*}H = \{1, 3 \} \quad \text{and} \quad K = \{1, 5 \}.\end{equation*}

Example9.25

The dihedral group \(D_6\) is an internal direct product of its two subgroups \begin{equation*}H = \{\identity, r^3 \} \quad \text{and} \quad K = \{\identity, r^2, r^4, s, r^2s, r^4 s \}.\end{equation*} It can easily be shown that \(K \cong S_3\); consequently, \(D_6 \cong {\mathbb Z}_2 \times S_3\).

Example9.26

Not every group can be written as the internal direct product of two of its proper subgroups. If the group \(S_3\) were an internal direct product of its proper subgroups \(H\) and \(K\), then one of the subgroups, say \(H\), would have to have order 3. In this case \(H\) is the subgroup \(\{ (1), (123), (132) \}\). The subgroup \(K\) must have order 2, but no matter which subgroup we choose for \(K\), the condition that \(hk = kh\) will never be satisfied for \(h \in H\) and \(k \in K\).

Proof
Example9.28

The group \({\mathbb Z}_6\) is an internal direct product isomorphic to \(\{ 0, 2, 4\} \times \{ 0, 3 \}\).

We can extend the definition of an internal direct product of \(G\) to a collection of subgroups \(H_1, H_2, \ldots, H_n\) of \(G\), by requiring that

  • \(G = H_1 H_2 \cdots H_n = \{ h_1 h_2 \cdots h_n : h_i \in H_i \}\);

  • \(H_i \cap \langle \cup_{j \neq i} H_j \rangle = \{ e \}\);

  • \(h_i h_j = h_j h_i\) for all \(h_i \in H_i\) and \(h_j \in H_j\).

We will leave the proof of the following theorem as an exercise.