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Direct sum

Definition

Given groups $G,H$ with binary operations $*_G,*_H$, the direct sum is the group $G \oplus H$ with elements $(g,h): g \in G, h \in H$, with the operation $(g_1,h_1)*(g_2,h_2)=(g_1*_Gg_2,h_1*_Hh_2)$.

Properties

$G\oplus H$ is abelian if an only if $G,H$ are both abelian.

If $g$ generates $G$ and $h$ generates $H$, then $(g,i_H), (i_G,h)$ together generate $G\oplus H$ (this can be generalised for groups that are generated by multiple elements).

all these pages adapted with probably insufficient credit from my university's lecture notes