### Bonus (10 points): Let ( G ) be an abelian group acting on a set ( X ). Suppose that there is only one orbit under this action. Assume that ( |G| neq|X| ). Prove ( ^{2} ) that there exists ( e neq g in G ) such that ( operatorname{Fix}(g)=X ).

Bonus (10 points): Let ( G ) be an abelian group acting on a set ( X ). Suppose that there is only one orbit under this action. Assume that ( |G| neq|X| ). Prove ( ^{2} ) that there exists ( e neq g in G ) such that ( operatorname{Fix}(g)=X ).