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Section 1.3 Direct product

We can also take products of groups to construct bigger groups. The natural operation is called direct product. This is analogous to the notion of direct sums of vector subspaces, where vectors are written as sums of vectors of different subspaces, such that the only common vector in the distinct subspaces is the zero vector.

Here we will first take the point of view where we study when a group is a direct product of two of its subgroups. We will then see how we can use the Cartesian product of two sets to naturally construct direct products of groups.

Definition 1.3.1. The direct product.

A group \(G\) is said to be the direct product of its subgroups \(H_1\) and \(H_2\text{,}\) which is denoted by \(G = H_1 \times H_2\text{,}\) if the following conditions are satisfied:

  1. All elements of \(H_1\) commute with all elements of \(H_2\text{;}\)
  2. The group identity is the only common element to \(H_1\) and \(H_2\text{;}\)
  3. Every \(g \in G\) can be written as \(g = h_1 h_2\) for some \(h_! \in H_1\) and \(h_2 \in H_2\text{.}\)

We can use this definition to construct bigger groups by taking direct products of groups:

Prove this proposition.

An example of a direct product is the group \(\mathbb{Z}_2 \times \mathbb{Z}_2\) (also called the “Klein four-group”). Using the construction above, one can think of this group as consisting of the four elements \((1,1), (1,-1), (-1,1), (-1,-1)\text{,}\) with the operation being component-wise multiplication. Note that this group is different from the cyclic group of order 4 \(\mathbb{Z}_4\text{;}\) for instance, in \(\mathbb{Z}_4 = \{1,-i,-1,i \}\) only two elements (\(1\) and \(-1\)) square to the identity, while in \(\mathbb{Z}_2 \times \mathbb{Z}_2\) all four elements square to the identity.

This notion of direct product is what is needed to make sense, for instance, of the gauge group of the Standard Model in particle physics, which is given by \(SU(3) \times SU(2) \times U(1)\text{.}\)

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