External Direct Products

External Direct Product

Let

G_{1}, G_{2}, \dots, G_{n}

be a finite collection of groups. The external direct product of

G_{1}, G_{2}, \dots, G_{n}

, written as

G_{1} \oplus︎ G_{2} \oplus︎ \dots \oplus︎ G_{n}

, is the set of all

n

-tuples for which the

i

th component is an element of

G_{i}

and the operation is componentwise. In symbols,

G_{1} \oplus︎ G_{2} \oplus︎ \dots \oplus︎ G_{n} = {(g_{1}, g_{2}, \dots, g_{n}) | g_{i} \in G_{i}}

The External Direct Product With Componentwise Multiplication Is a Group

Let

G_{1} \oplus︎ G_{2} \oplus︎ \dots \oplus︎ G_{n}

be a direct propduct, where we define multiplication as

(g_{1}, g_{2}, \dots, g_{n}) (g_{1}^{'}, g_{2}^{'}, \dots, g_{n}^{'}) = (g_{1} g_{1}^{'}, g_{2} g_{2}^{'}, \dots, g_{n} g_{n}^{'})

It is understood that each product

g_{i} g_{i}^{'}

is performed with the operation of

G_{i}

, then this forms a group

TODO: Add the proof here.

Note that in the case that each $G_{i}$ is finite, we have by properties of sets that $| G_{1} \oplus︎ G_{2} \oplus︎ \dots \oplus︎ G_{n} | = | G_{1} | | G_{2} | \dots | G_{n} |$ .