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If $\\sA$ and $\\sB$ are groups of $G$-valued maps defined on the sets $X$ and $Y$, respectively, we say that $\\sA$ and $\\sB$ are \\emph{equivalent} if there is a group isomorphism $H\\colon\\sA\\to\\sB$ such that there is a bijective map $h\\colon Y\\to X$ and a map $w\\colon Y\\to \\sA ut (G)$ satisfying $Hf(y)=w[y](f(h(y)))$ for all $y\\in Y$ and $f\\in \\sA$. In this case, we say that $H$ is represented as a \\emph{weighted composition operator}. 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