Multicoset for a tuple of subgroups

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Definition

Let $G$ be a group and $H_{1},H_{2},\ldots ,H_{r}$ be an $r$-tuple of subgroups of $G$. Consider the natural action of $G$ on the coset space $G/H_{1}\times G/H_{2}\times \ldots G/H_{r}$. The orbits under this action are termed the multicosets for the tuple of subgroups.

Importance

Multicoset generalizes the notion of left coset and the more general notion of double coset, to a tuple of $r$ subgroups for $r>2$.