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If every element of a $G$-set is left fixed by the same element $g$ of $G$, then $g$ must be the identity $e$.

I believe this to be true, but the answers say that it's false. Can anyone provide a counter-example? Thanks!

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    By a $G$-set, do you mean a set on which $G$ acts? In that case, you can let the set have one element and let all elements of $G$ fix that element.2012-01-29

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