F1 - definition

external direct product

let $G_{1}, G_{2}, \dots, G_{n}$ be a finite collection of groups, then their external direct product, written as $G_{1} \oplus G_{2} \oplus \dots \oplus G_{n}$ , is the set of all $n$ -tuples for which the $i$ th component is an element of $G_{i}$ , and the operation is component-wise

\oplus_{i = 1}^{n} G_{i} = G_{1} \oplus G_{2} \oplus \dots \oplus G_{n} = {(g_{1}, g_{2}, \dots, g_{n}) ∣ g_{i} \in G_{i}}

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

each product $g_{i} g_{i}^{'}$ is performed with the operation of $G_{i}$
if each $G_{i}$ is finite, $| \oplus_{i = 1}^{n} G_{i} | = \sum_{i = 1}^{n} | G_{i} |$
the external direct product of groups is a group
eg: $R^{2} = R \oplus R$ and $R^{3} = R \oplus R \oplus R$
eg:

\begin{matrix} U (8) \oplus U (10) = & {(1, 1), (1, 3), (1, 7), (1, 9), \\ (3, 1), (3, 3), (3, 7), (3, 9), \\ (5, 1), (5, 3), (5, 7), (5, 9), \\ (7, 1), (7, 3), (7, 7), (7, 9)} \end{matrix}

the product $(3, 7) (7, 9) = (5, 3)$ as the first components are combined by multiplication modulo 8, whereas the second with 10
in some cases, $Z_{m} \oplus Z_{N} ≅ Z_{m n}$