Section 11.3 Direct Products
Given two groups \(G\) and \(H\text{,}\) it is possible to construct a new group from the Cartesian product of \(G\) and \(H\text{,}\) \(G \times H\text{.}\) 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 \(G\text{,}\) it is often easier to study the component groups of \(G\text{.}\)
Worksheet 11.3.1 Activity: Decomposing with Direct Products
Recall that last semester we saw that \(\Z_6 \cong \Z_2\times \Z_3\text{.}\) When does this sort of thing happen?
1.
Given positive integers \(m\) and \(n\text{,}\) is it always true that \(\Z_{mn} \cong \Z_m\times \Z_n\text{?}\) If this is not always true, for which \(m\) and \(n\) is it true? Try some (many) examples.
2.
Consider \(\Z_{12}\text{.}\) Can we break this down as the direct product of two smaller \(\Z_p\) groups? In other words is \(\Z_{12} = \Z_m \times \Z_n\) for some values of \(m\) and \(n\text{?}\)
3.
Suppose your absent minded professor claims the answer is “no” and you don't feel like arguing. Maybe we can do something similar. Find two subgroups of \(\Z_{12}\text{,}\) call them \(H\) and \(K\text{,}\) such that \(H \cap K = \{0\}\) and \(HK = \Z_{12}\text{.}\) In general, \(HK = \{h\ast k \st h \in H, k \in K\}\text{;}\) here it would be better to write \(H+K\text{.}\)
For any \(n\text{,}\) the group \(U(n)\) is the set of all positive integers less than and relatively prime to \(n\text{,}\) under multiplication modulo \(n\text{.}\) For example we saw that \(U(8) = \{1,3, 5, 7\}\) is a group under multiplication modulo 8.
Consider the group \(U(28)\text{.}\) The table below gives the twelve elements with their orders:
| \(g\) | 1 | 3 | 5 | 9 | 11 | 13 | 15 | 17 | 19 | 23 | 25 | 27 |
| \(\mathrm{ord}(g)\) | 1 | 6 | 6 | 3 | 6 | 2 | 2 | 6 | 6 | 6 | 3 | 2 |
4.
Let \(G(n)\) be the set of all elements of order \(n^k\) for some \(k\) (that is, elements with order some power of \(n\)). Find \(G(2)\) and \(G(3)\) for \(U(28)\text{.}\)
5.
Are \(G(2)\) and \(G(3)\) subgroups of \(U(28)\text{?}\)
6.
Do \(G(2)\) and \(G(3)\) have the property that \(G(2) \cap G(3) = \{1\}\) and \(U(28) = G(2)G(3)\text{?}\)
7.
Is \(U(28) \cong G(2) \times G(3)\text{?}\) Is \(U(28) \cong \Z_m\times \Z_n\) for some values of \(m\) and \(n\text{?}\)
Subsection 11.3.2 External 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 \(h \in H\text{.}\) We can define a binary operation on \(G \times H\) by
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 \(H\text{.}\) We have specified the particular operations \(\cdot\) and \(\circ\) in each group here for the sake of clarity; we usually just write \((g_1, h_1)(g_2, h_2) = (g_1 g_2, h_1 h_2)\text{.}\)
Proposition 11.27.
Let \(G\) and \(H\) be groups. The set \(G \times H\) is a group under the operation \((g_1, h_1)(g_2, h_2) = (g_1 g_2, h_1 h_2)\) where \(g_1, g_2 \in G\) and \(h_1, h_2 \in H\text{.}\)
Proof.
Clearly the binary operation defined above is closed. If \(e_G\) and \(e_H\) are the identities of the groups \(G\) and \(H\) respectively, then \((e_G, e_H)\) is the identity of \(G \times H\text{.}\) The inverse of \((g, h) \in G \times H\) is \((g^{-1}, h^{-1})\text{.}\) The fact that the operation is associative follows directly from the associativity of \(G\) and \(H\text{.}\)
Example 11.28.
Let \({\mathbb R}\) be the group of real numbers under addition. The Cartesian product of \({\mathbb R}\) with itself, \({\mathbb R} \times {\mathbb R} = {\mathbb R}^2\text{,}\) is also a group, in which the group operation is just addition in each coordinate; that is, \((a, b) + (c, d) = (a + c, b + d)\text{.}\) The identity is \((0,0)\) and the inverse of \((a, b)\) is \((-a, -b)\text{.}\)
Example 11.29.
Consider
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 \(2\text{,}\) since \((a,b) + (a,b) = (0,0)\text{;}\) however, \({\mathbb Z}_4\) is cyclic.
The group \(G \times H\) is called the external direct product of \(G\) and \(H\text{.}\) Notice that there is nothing special about the fact that we have used only two groups to build a new group. The direct product
of the groups \(G_1, G_2, \ldots, G_n\) is defined in exactly the same manner. If \(G = G_1 = G_2 = \cdots = G_n\text{,}\) we often write \(G^n\) instead of \(G_1 \times G_2 \times \cdots \times G_n\text{.}\)
Example 11.30.
The group \({\mathbb Z}_2^n\text{,}\) 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,
This group is important in coding theory, in cryptography, and in many areas of computer science.
Theorem 11.31.
Let \((g, h) \in G \times H\text{.}\) If \(g\) and \(h\) have finite orders \(r\) and \(s\) respectively, then the order of \((g, h)\) in \(G \times H\) is the least common multiple of \(r\) and \(s\text{.}\)
Proof.
Suppose that \(m\) is the least common multiple of \(r\) and \(s\) and let \(n = |(g,h)|\text{.}\) Then
Hence, \(n\) must divide \(m\text{,}\) and \(n \leq m\text{.}\) However, by the second equation, both \(r\) and \(s\) must divide \(n\text{;}\) therefore, \(n\) is a common multiple of \(r\) and \(s\text{.}\) Since \(m\) is the least common multiple of \(r\) and \(s\text{,}\) \(m \leq n\text{.}\) Consequently, \(m\) must be equal to \(n\text{.}\)
Corollary 11.32.
Let \((g_1, \ldots, g_n) \in \prod G_i\text{.}\) If \(g_i\) has finite order \(r_i\) in \(G_i\text{,}\) then the order of \((g_1, \ldots, g_n)\) in \(\prod G_i\) is the least common multiple of \(r_1, \ldots, r_n\text{.}\)
Example 11.33.
Let \((8, 56) \in {\mathbb Z}_{12} \times {\mathbb Z}_{60}\text{.}\) Since \(\gcd(8,12) = 4\text{,}\) the order of \(8\) is \(12/4 = 3\) in \({\mathbb Z}_{12}\text{.}\) Similarly, the order of \(56\) in \({\mathbb Z}_{60}\) is \(15\text{.}\) The least common multiple of \(3\) and \(15\) is \(15\text{;}\) hence, \((8, 56)\) has order \(15\) in \({\mathbb Z}_{12} \times {\mathbb Z}_{60}\text{.}\)
Example 11.34.
The group \({\mathbb Z}_2 \times {\mathbb Z}_3\) consists of the pairs
In this case, unlike that of \({\mathbb Z}_2 \times {\mathbb Z}_2\) and \({\mathbb Z}_4\text{,}\) it is true that \({\mathbb Z}_2 \times {\mathbb Z}_3 \cong {\mathbb Z}_6\text{.}\) 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 \({\mathbb Z}_2 \times {\mathbb Z}_3\text{.}\)
The next theorem tells us exactly when the direct product of two cyclic groups is cyclic.
Theorem 11.35.
The group \({\mathbb Z}_m \times {\mathbb Z}_n\) is isomorphic to \({\mathbb Z}_{mn}\) if and only if \(\gcd(m,n)=1\text{.}\)
Proof.
We will first show that if \({\mathbb Z}_m \times {\mathbb Z}_n \cong {\mathbb Z}_{mn}\text{,}\) then \(\gcd(m, n) = 1\text{.}\) We will prove the contrapositive; that is, we will show that if \(\gcd(m, n) = d \gt 1\text{,}\) then \({\mathbb Z}_m \times {\mathbb Z}_n\) cannot be cyclic. Notice that \(mn/d\) is divisible by both \(m\) and \(n\text{;}\) hence, for any element \((a,b) \in {\mathbb Z}_m \times {\mathbb Z}_n\text{,}\)
Therefore, no \((a, b)\) can generate all of \({\mathbb Z}_m \times {\mathbb Z}_n\text{.}\)
The converse follows directly from Theorem 11.31 since \(\lcm(m,n) = mn\) if and only if \(\gcd(m,n)=1\text{.}\)
Corollary 11.36.
Let \(n_1, \ldots, n_k\) be positive integers. Then
if and only if \(\gcd( n_i, n_j) =1\) for \(i \neq j\text{.}\)
Corollary 11.37.
If
where the \(p_i\)s are distinct primes, then
Proof.
Since the greatest common divisor of \(p_i^{e_i}\) and \(p_j^{e_j}\) is 1 for \(i \neq j\text{,}\) the proof follows from Corollary 11.36.
In Section 11.4, we will prove that all finite abelian groups are isomorphic to direct products of the form
where \(p_1, \ldots, p_k\) are (not necessarily distinct) primes.
Subsection 11.3.3 Internal 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.
\(G = HK = \{ hk : h \in H, k \in K \}\text{;}\)
\(H \cap K = \{ e \}\text{;}\)
\(hk = kh\) for all \(k \in K\) and \(h \in H\text{.}\)
Then \(G\) is the internal direct product of \(H\) and \(K\text{.}\)
Example 11.38.
The group \(U(8)\) is the internal direct product of
Example 11.39.
The dihedral group \(D_6\) is an internal direct product of its two subgroups
It can easily be shown that \(K \cong S_3\text{;}\) consequently, \(D_6 \cong {\mathbb Z}_2 \times S_3\text{.}\)
Example 11.40.
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 \(K\text{,}\) then one of the subgroups, say \(H\text{,}\) would have to have order \(3\text{.}\) In this case \(H\) is the subgroup \(\{ (1), (123), (132) \}\text{.}\) The subgroup \(K\) must have order \(2\text{,}\) but no matter which subgroup we choose for \(K\text{,}\) the condition that \(hk = kh\) will never be satisfied for \(h \in H\) and \(k \in K\text{.}\)
Theorem 11.41.
Let \(G\) be the internal direct product of subgroups \(H\) and \(K\text{.}\) Then \(G\) is isomorphic to \(H \times K\text{.}\)
Proof.
Since \(G\) is an internal direct product, we can write any element \(g \in G\) as \(g =hk\) for some \(h \in H\) and some \(k \in K\text{.}\) Define a map \(\phi : G \rightarrow H \times K\) by \(\phi(g) = (h,k)\text{.}\)
The first problem that we must face is to show that \(\phi\) is a well-defined map; that is, we must show that \(h\) and \(k\) are uniquely determined by \(g\text{.}\) Suppose that \(g = hk=h'k'\text{.}\) Then \(h^{-1} h'= k (k')^{-1}\) is in both \(H\) and \(K\text{,}\) so it must be the identity. Therefore, \(h = h'\) and \(k = k'\text{,}\) which proves that \(\phi\) is, indeed, well-defined.
To show that \(\phi\) preserves the group operation, let \(g_1 = h_1 k_1\) and \(g_2 = h_2 k_2\) and observe that
We will leave the proof that \(\phi\) is one-to-one and onto as an exercise.
Example 11.42.
The group \({\mathbb Z}_6\) is an internal direct product isomorphic to \(\{ 0, 2, 4\} \times \{ 0, 3 \}\text{.}\)
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 \(G\text{,}\) by requiring that
\(G = H_1 H_2 \cdots H_n = \{ h_1 h_2 \cdots h_n : h_i \in H_i \}\text{;}\)
\(H_i \cap \langle \cup_{j \neq i} H_j \rangle = \{ e \}\text{;}\)
\(h_i h_j = h_j h_i\) for all \(h_i \in H_i\) and \(h_j \in H_j\text{.}\)
We will leave the proof of the following theorem as an exercise.
Theorem 11.43.
Let \(G\) be the internal direct product of subgroups \(H_i\text{,}\) where \(i = 1, 2, \ldots, n\text{.}\) Then \(G\) is isomorphic to \(\prod_i H_i\text{.}\)
Exercises 11.3.4 Collected Homework
1.
Consider the group \(U(35) = \{1,2,3,4,6,8,9,11,12,13,16,17,18,19,22,23,24,26,27,29,31,32,33,34\}\) under the operation of multiplication modulo \(35\text{.}\) The orders of the elements are:
| \(g\) | 1 | 2 | 3 | 4 | 6 | 8 | 9 | 11 | 12 | 13 | 16 | 17 | 18 | 19 | 22 | 23 | 24 | 26 | 27 | 29 | 31 | 32 | 33 | 34 |
| \(\ord(g)\) | 1 | 12 | 12 | 6 | 2 | 4 | 6 | 3 | 12 | 4 | 3 | 12 | 12 | 6 | 4 | 12 | 6 | 6 | 4 | 2 | 6 | 12 | 12 | 2 |
Find two \(p\)-groups \(H\) and \(K\) such that \(U(35)\) is the internal direct product of \(H\) and \(K\text{.}\) Briefly explain why your groups work.
Let \(H\) be the larger of the two groups above. Show how to decompose it as the internal direct product of \(\langle a \rangle\) and \(H'\) where \(a\) is of maximal order and \(H'\) is some other subgroup of \(H\text{.}\)
Using the decompositions above (perhaps repeating the second step as needed), write \(U(35)\) as the direct product of groups of the form \(\Z_{p^k}\) (\(p\) prime).