We know that for a cyclic group $G$, if $G=A\oplus B$, then for some subgroups $H$ of $G$. We have $G/H=A/H\oplus B/H.$ But, if we know that for a subgroup $H$ of $G$, $G/H=A/H\oplus B/H.$ Then what conditions do we need to make $G=A\oplus B$?
From direct sum of quotient group of a group to direct sum of the group
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