1
$\begingroup$

Let $G$ be a transitive subgroup of $S_n$ generated by a transposition and a cycle of order $p$ whith $p$ a prime and $\frac{n}{2} < p < n$. Prove that $G=S_n$. Please ,I would like for the moment just a hint to tackle the problem.

$\endgroup$
1
$\begingroup$

We have to show that the group operation for $G$ is an equivalence relation, and the main thing is to show is that the number of equivalence classes is $1$. Since we have

Order of $G$ =(size of an equivalence class, say $M$)$\times$(number of equivalence class, say $N$).

i.e. $n=MN$. Now, we must show that size of any equivalence class is same and $N=1$. For the latter half, one can see whether an orbit of $G$ with size $p$ exists or not.

Or, alternatively,

take a look on page 2 and page 4.

http://www.math.uconn.edu/~kconrad/blurbs/galoistheory/galoisSnAn.pdf

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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