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Thread: G-set

  1. #1
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    G-set

    "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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  2. #2
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    Quote Originally Posted by aliceinwonderland View Post
    "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 $\displaystyle G$ be any group and let $\displaystyle X=G.$ define the action by $\displaystyle g \cdot x=gxg^{-1}, \ \forall g \in G, \ \forall x \in X.$ now if $\displaystyle G$ is a non-trivial abelian group, then $\displaystyle g \cdot x=x, \ \forall g \in G, \ \forall x \in X.$
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