Direct Products of Groups

Componentwise products, element orders, and the recognition theorem.

The idea

The direct product is the simplest way to combine two groups $G$ and $H$ into one, and the simplest way to break a large group into smaller ones.

Definition (Direct product).

For groups $G$ and $H$, the direct product $G \times H$ is the set of ordered pairs $(g, h)$ with $g \in G$ and $h \in H$, multiplied coordinate by coordinate: $(g_1, h_1)(g_2, h_2) = (g_1 g_2,\ h_1 h_2).$

The identity is $(e_G, e_H)$ and the inverse is $(g, h)^{-1} = (g^{-1}, h^{-1})$; the group axioms hold because they hold in each coordinate separately. Built this way, from two groups given in advance, $G \times H$ is called the external direct product; the two coordinates never interact.

Because the coordinates are independent, questions about $G \times H$ reduce to the same question asked in $G$ and in $H$. Counting pairs gives $|G \times H| = |G|\,|H|$. Powers are computed coordinatewise, $(g, h)^{k} = (g^{k}, h^{k})$, so a power is the identity exactly when both coordinates reach their identities, and this determines the order of every element of the product. In $\mathbb{Z}_m \times \mathbb{Z}_n$, for instance, the powers of $(1, 1)$ step diagonally through the grid of the $mn$ pairs, wrapping past each edge, and first return to $(0, 0)$ after $\text{lcm}(m, n)$ steps.

The construction also runs in reverse: a group may already contain two subgroups from which it is built as a product, and recognizing this reduces its study to the study of two smaller groups.

Theorem (Recognition of direct products).

Let $A$ and $B$ be normal subgroups of a group $K$ with $A \cap B = \{e\}$ and $AB = K$. Then $K \cong A \times B$, and $K$ is called the internal direct product of $A$ and $B$.

The conditions force every element of $K$ to factor uniquely as $ab$ with $a \in A$ and $b \in B$, with elements of $A$ commuting with elements of $B$; the isomorphism sends $ab$ to $(a, b)$.

Ways to work on it

• Walkthrough. Order of a product, order of an element, and the recognition theorem.
• Practice. Find the order of an element in a direct product.
• Hardest. Decide whether a product of cyclic groups is itself cyclic.

Not sure where to start? Take the ten-question placement test.