Direct 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 \(G\).

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 \[\begin{aligned}\end{aligned}\] 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 \((g_1, h_1)(g_2, h_2) = (g_1 g_2, h_1 h_2)\).

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 \[\begin{aligned}\end{aligned}\] 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 \(G_1 \times G_2 \times \cdots \times G_n\).

Condensed: the full section is in Judson, Abstract Algebra: Theory and Applications.

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 \(h \in H\).

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 \(h_j \in H_j\).

Sage can quickly determine if two permutation groups are isomorphic, even though this should, in theory, be a very difficult computation.

Condensed: the full section is in Judson, Abstract Algebra: Theory and Applications.