Good afternoon fellow math goers.

I entered this problem a week ago and with helpful hints have made what I think is progression on this rather challenging problem. I need some advice on my proof though so any more help would be greatly appreciated.

Problem: Prove that the smallest subgroup of Sn containing (1, 2) and (1, 2, . . . , n) is Sn.

Conjunction:

(12...n)(12)(12...n)^-1=(23)

(12...n)(23)(12...n)^1=(34)

(12...n)(34)(12...n)^-1=(45)

WLOG (12...n)(n-2 n) (12...n)^-1=(n-1 n)

(12...n)^n-1 (n-1 n)(12...n)^n-1=(n1)

So essentially this gives us a difference k=0 and that we proved its a generator of Sn correct? Now should I use induction and do k+1 terms or should I do the following:

Suppose you have some group such that ,

Above I proved that (12)(1...n) generates Sn so therefore

How do I prove that . so I can establish that G=Sn??

AHHHH