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Physics

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    $\begingroup$ I read your condition (2) as "given an element" as "for every element $e \in V$" – then the two statements are not equivalent, as $0 \in V$ and $\{R_g(0) \colon g \in G \}$ is the null vector space. $\endgroup$
    – Sebastian Riese
    Commented Apr 16 at 13:38
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    $\begingroup$ You are right, I edited the second statement so as to be more precise $\endgroup$
    – Gippo
    Commented Apr 16 at 16:12