Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

How come a transitive subgroup of $S_5$ must have order divisible by 5? I can see why any subgroup of $S_5$ with order divisible by 5 is transitive, but I do not see the reverse.

share|improve this question
add comment

1 Answer

The orbit-stabilizer theorem tells us that, if a finite group $G$ acts on a finite set $X$, then, for any $x$ in $X$, the order of $G$ is equal to the order of the stabilizer of $x$ times the size of the orbit of $x$. If the action is transitive, then every orbit is $X$ itself, so we have $|G|=|\mathrm{Stab}(x)||X|$, and so $|X|$ divides $|G|$. Applying this to the natural action of our subgroup on $\{1,2,3,4,5\}$ gives the result.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.