Group theory/Define product group/Mappings/Exercise

Let ${}G_{1},\ldots ,G_{n}$ be groups.

a) Define a group structure on the product

$G_{1}\times \cdots \times G_{n}.$

b) Let ${}H$ be another group. Show that a mapping

$\varphi \colon H\longrightarrow G_{1}\times \cdots \times G_{n},x\longmapsto \varphi (x)=(\varphi _{1}(x),\ldots ,\varphi _{n}(x)),$

is a group homomorphism if and only if all components ${}\varphi _{i}$ are group homomorphisms.

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