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The group structure of G plays no role in the definition of a G-module. For any set ${\displaystyle \Omega }$, D.J.S. Robinson, A Course in the Theory of Groups, 2d Edition, p. 131, speaks of a ${\displaystyle \Omega }$-module as being an abelian group with operator domain ${\displaystyle \Omega }$. Perhaps It might be better to start by defining a ${\displaystyle \Omega }$-module for a set ${\displaystyle \Omega }$ and then say that a G-module is exactly a ${\displaystyle \Omega }$-module with ${\displaystyle \Omega }$ = G ? Marvoir ( talk) 07:48, 10 August 2021 (UTC) reply