2

I have the following problem: Let $G$ be a finite group with transitive group action on a set $X$ (#$X > 1$). Prove: there exists an element $g \in G$ such that $gx \neq x$ for all $x \in X$.

It would be nice if anyone could help me getting started. Usually with these problems, I have trouble understanding what information I have given. I know that transitive action is: $Gx = X$ for some $x \in X$. But how can I use this definition?

Regards.

Katie
  • 437
  • If it's not the case then $\displaystyle\bigcup_{x\in X}Stab(x) = G$. You can use the orbit-stabiliser theorem to compute the order of $Stab(x)$ and find a contradiction – Maxime Ramzi Apr 24 '18 at 10:38
  • This question was already asked, you can find the answer here: https://math.stackexchange.com/questions/1820572/g-acts-on-x-transitively-then-there-exists-some-element-that-does-not-have-any – M.Pintonello Apr 24 '18 at 10:41
  • A hint to get started is "Burnsides lemma". If you want a full solution, click on the above link. – user1729 Apr 24 '18 at 11:03

0 Answers0