G-equidecomposable and equivalence relation

AI Thread Summary
The discussion centers on proving that the relation A~B is an equivalence relation on subsets of X if and only if A and B are G-equidecomposable. Participants note that this concept relates to the Banach-Schroder theorem and involves checking three properties of equidecomposability, which are generally straightforward due to group properties like inverses and associativity. The definition of G-equidecomposable is requested for clarity, indicating a need for foundational understanding. The proof is suggested to be simple, relying on the established properties of groups. Overall, the conversation emphasizes the relationship between group actions and the equivalence of subsets through G-equidecomposability.
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G acts via isometries on a set X, and A,B are subsets of X. Prove that the relation A~B is an equivalence relation on subsets of X iff A and B are G-equidecomposable.


I think this has to do with the Banach-Schroder theorem, but am not sure. I know it is a definition in group theory, but am not sure how to prove it since it seem pretty self explanitory to me.
 
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Can someone help?
 
define G-equidecomposable, please,

the result should be easy isnce it just checking 3 things for the property of equidecomposability, whatever that may be, which are generally trivial for groups since groups have inverses, and identity and composition is associative. eg if X~Y and g in G effects this relation, then g^{-1} will (probably) effect the relation Y~X, if X~Y and Y~Z and the relation is because of elements f and g resp. then gf wil mean that X~Y, and X~X because e(X)=X.

note i haven't a clue what equidecompsoability is, but this will be the proof, I'm almost sure of it.
 

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