= Equal-or-disjoint translates characterize cosets
{title2=$gB\cap hB\ne\varnothing\Longrightarrow gB=hB$}
A nonempty <subset> $B$ of any <group> $G$ is a <left coset> of a <subgroup> if and only if any two left translates $gB,hB$ are equal or disjoint. No finiteness assumption is needed.
For the forward direction, write $B=bH$ with $H$ a <subgroup>. Then $gB=gbH$, and distinct <left cosets> of $H$ are disjoint. For the converse, first assume $e\in B$. For every $b\in B$, $b\in B\cap bB$, so $bB=B$. This proves closure under multiplication. Since $e\in B=bB$, there is $c\in B$ with $bc=e$, showing $b^{-1}\in B$. Thus $B$ passes the <subgroup test>.
For a general nonempty $B$, choose $b\in B$ and set $H=b^{-1}B$. It contains $e$, and all its left translates have the assumed property because $gH=(gb^{-1})B$. The previous argument makes $H$ a <subgroup>, and $B=bH$ is a <left coset>. This gives a recognition criterion for <cosets> from their translate intersections alone.
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