= Restricted direct sum of groups
{title2=$\bigoplus_iG_i$}
The restricted <direct sum> of a family of <groups> $G_i$ is the <subgroup> of their <direct product of groups> consisting of tuples with only finitely many nonidentity coordinates. Multiplication is coordinatewise. If every factor is a <finite group>, each <subgroup> that is a <finitely generated group> lies in a finite subproduct, so the restricted <direct sum> is a <locally finite group>. This gives concrete infinite <torsion groups> without a uniform bound on element orders.
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