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Let $G$ be a group, $\beta G$ be the Stone-Čech compactification of $G$ endowed with the structure of a right topological semigroup and $G^{*}=\beta G\setminus G$. Given any subset $A$ of $G$ and $p\in G^{*}$, we define the $p$-companion ${\Delta }_p\left(A\right)=A^{*}\cap Gp$ of $A$, and characterize the subsets with finite and discrete ultracompanions.