# permutation/transitivity

• November 13th 2009, 07:58 PM
apple2009
permutation/transitivity
Prove: Let H be a finite group of permutations of the set Y. Suppose Y acts transitivity on Y. Then Y is a finite set, and |Y|divide |H|
• November 13th 2009, 08:55 PM
Jose27
It's a standard theorem that $H/St_H(x) \equiv orb(x)$ as H-sets where $St_H(x):= \{ h \in H : hx=x \}$ and $orb(x)= \{ hx \in Y : h\in H \}$ and since $H$ acts transitively $orb(x)=Y$ so we have $\vert H \vert = \vert orb(x) \vert \vert St_H(x) \vert = \vert Y \vert \vert St_H(x) \vert$ and the result is immediate.