# Math Help - G-set

1. ## 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?

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