If one says "Suppose $H\leq G$, and let $X$ be the $G$-set $G/H$ ...", from the word ${\it the}$, is it implied that the group action here is left multiplication/left translation?
2026-03-30 08:15:01.1774858501
canonical action of G acting on its subsets
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Yes. $G/H$ is the set of left-cosets of $G$ by $H$, so the implied action in this case is given by $g\cdot (g'H) = (gg')H$.