A Functional Treatment of the Direct Product of Groups

Definition of direct products

Let \(A\) be an arbitrary well-ordered set and \(\{ G_{\alpha} \vert \alpha \in A \}\) a set of groups under the operations \(\{ *_{\alpha} \vert \alpha \in A \}\) respectively,

\[\forall \: \alpha \in A : *_{\alpha} \in \left( G_{\alpha} \right)^{G_{\alpha} \times G_{\alpha}}\]

One way to define the direct product of these groups is via the standard Cartesian product,

The direct product of the entries of \((G_{\alpha} \vert \alpha \in A)\) for well-ordered set \(A\) is the set,

\[\prod_{\alpha \in A} G_\alpha = \left\{ \left( a_{\alpha} \right)_{\alpha \in A} \middle\vert \forall \: \alpha \in A : a_{\alpha} \in G_{\alpha} \right\}\]

equipped with the symmetric group operation \(\displaystyle{* : \prod_{\alpha \in A} G_\alpha \times \prod_{\alpha \in A} G_\alpha \to \prod_{\alpha \in A} G_\alpha}\),

\[\forall \: a, b \in \prod_{\alpha \in A} G_\alpha : a * b = (a_{\alpha} *_{\alpha} b_{\alpha} \vert \alpha \in A)\]

Isomorphisms

Let \(\cong_{\text{Set}}\) denote set-theoretic isomorphism i.e. isomorphism in the category \(\text{Set}\), which is equivalence of sets under bijective maps.¹ In general, let \(\cong_C\) denote isomorphism in a small category \(C\) i.e. one in the category \(\text{Cat}\).

Then, in the category \(\text{Grp}\) of groups, by definition,

\[(G, *_{G}) \cong_{\text{Grp}} (H, *_{H}) \iff \left[ G \overset{\varphi : G \to H}{\cong_{\text{Set}}} H \right] \land \left[ \forall \: u, v \in G : \varphi(u *_{G} v) = \varphi(u) *_{H} \varphi(v) \right]\]

Functional construction of direct products

Statement

We want to prove that the group-theoretic direct product is group-isomorphic to the following one,

\[G = \left\{ f \in \left( \bigcup_{\alpha \in A} G_{\alpha} \right)^{A} \middle\vert \forall \: \alpha \in A : f(\alpha) \in G_{\alpha} \right\}\]

when equipped with the binary operation \(\displaystyle{*_{G}} : \left( \bigcup_{\alpha \in A} G_{\alpha} \right)^{A} \times \left( \bigcup_{\alpha \in A} G_{\alpha} \right)^{A} \to \left( \bigcup_{\alpha \in A} G_{\alpha} \right)^{A}\),

\[\forall \: f, g \in G : f *_{G} g = f \circ g\]

i.e.,

\[\left( \prod_{\alpha \in A} G_{\alpha}, * \right) \cong_{\text{Grp}} (G, *_{G})\]

Proof

The conditions for the group isomorphism of the above groups are satisfied as follows:

• As per the definition of direct products,

\[\begin{align*} \prod_{\alpha \in A} G_\alpha & = \left\{ (a_{\alpha})_{\alpha \in A} \middle\vert \forall \: \alpha \in A : a_{\alpha} \in G_{\alpha} \right\} \\ & = \left\{ (f(\alpha))_{\alpha \in A} \middle\vert f \in \left( \bigcup_{\alpha \in A} G_{\alpha} \right)^{A}, \forall \: \alpha \in A : f(\alpha) \in G_{\alpha} \right\} \\ & \overset{\varphi}{\cong_{\text{Set}}} \left\{ \text{im}_{f}(A) \middle\vert f \in \left(\bigcup_{\alpha \in A} G_{\alpha} \right)^{A}, \forall \: \alpha \in A : f(\alpha) \in G_{\alpha} \right\} \\ & \overset{\eta}{\cong_{\text{Set}}} \left\{ f \middle\vert f \in \left( \bigcup_{\alpha \in A} G_{\alpha} \right)^{A}, \forall \: \alpha \in A : f(\alpha) \in G_{\alpha} \right\} \\ & = G \end{align*}\]

where \(\displaystyle{\varphi : \prod_{\alpha \in A} G_{\alpha} \to \bigcup_{\alpha \in A} G_{\alpha}}\) is the bijection,

\[\forall \: U \in \prod_{\alpha \in A} G_{\alpha} : \varphi(U) = \set{U_{\alpha}}{\alpha \in A}\]

with the inverse \(\displaystyle{\varphi^{-1} : \bigcup_{\alpha \in A} G_{\alpha} \to \prod_{\alpha \in A} G_{\alpha}}\),

\[\forall \: V \in \bigcup_{\alpha \in A} G_{\alpha} : \varphi^{-1}(V) = \left( \alpha \in V \vert \alpha \in V \right)\]

Similarly \(\displaystyle{\eta : }\) is the bijection \(\displaystyle{\bigcup_{\alpha \in A} G_{\alpha} \to \left( \bigcup_{\alpha \in A} G_{\alpha} \right)^A}\),

\[\forall \: U \in \bigcup_{\alpha \in A} G_{\alpha} : \eta(U) = \left\{ (\text{preim}_{f}(a), a) \middle\vert a \in U \right\}\]

with the inverse \(\displaystyle{\eta^{-1} : \left(\bigcup_{\alpha \in A} G_{\alpha} \right)^A \to \bigcup_{\alpha \in A} G_{\alpha}}\),

\[\forall \: V \in \left( \bigcup_{\alpha \in A} G_{\alpha} \right)^A : \eta^{-1}(V) = \text{im}_{f}(V)\]

It can be verified that the composition of the above maps, i.e. is the bijection \(\displaystyle{\eta \circ \varphi : \prod_{\alpha \in A} G_{\alpha} \to G}\),

\[\forall \: a \in \prod_{\alpha \in A} G_{\alpha} : (\eta \circ \varphi)(a) = \eta(\varphi(a)) = \left\{ (\alpha, a_{\alpha}) \middle\vert \alpha \in A \right\}\]

with the inverse \((\eta \circ \varphi)^{-1} = \varphi^{-1} \circ \eta^{-1}\).

• Using the definition of direct products and the condition for group isomorphism,

\[\begin{align*} \forall \: a, b \in \prod_{\alpha \in A} G_{\alpha} : (\eta \circ \varphi) (a * b) & = (\eta \circ \varphi) ((a_{\alpha} *_{\alpha} b_{\alpha} \vert \alpha \in A)) \\ & = \{ (\alpha, a_{\alpha} *_{\alpha} b_{\alpha}) \vert \alpha \in A \} \\ & = \{ (\alpha, a_{\alpha}) \vert \alpha \in A \} *_{G} \{ (\alpha, b_{\alpha}) \vert \alpha \in A \} \\ & = (\eta \circ \varphi)(a) *_{G} (\eta \circ \varphi) (b) \end{align*}\]

as required.

1. in the plural, as if \(f \in B^A\) is a bijection, so are all the elements in \(\displaystyle{\left\{ \underset{{f \in U}}{\bigcirc} f \middle\vert U \in \mathcal{P} \left( B^A \right) \right\}}\) where \(\bigcirc\) is recursive composition. For an infinite set of functions, one would use transfinite composition in the sense of category theory. ↩