"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 think the above statement is true, but the answer is false. Why the above one is a false statement? Any counterexample?

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Originally Posted by aliceinwonderland "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 think the above statement is true, but the answer is false. Why the above one is a false statement? Any counterexample? let be any group and let define the action by now if is a non-trivial abelian group, then

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