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i have a question

what are the condition on the function $f$ ?

so that this equality hold :

$ \sum_{x\in G} f(x) = \sum_{x\in f(G)} x $

is $f$ surjective is a necessary for this question ?

please help me with this question

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  • $\begingroup$ Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. $\endgroup$
    – Community Bot
    Commented Jan 21 at 12:03

1 Answer 1

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Injectivity is necessary and sufficient (unless $f=0$ at all points that appear multiple times), as in the left sum, you sum up over all elements in the image WITH multiplicity, so you count them multiple times if there are different $a,b$ s.t. $f(a)=f(b)$ and on the right hand side you just sum up over all elements in the image WITHOUT multiplicity.

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