
Smallest Subgroup of Sn
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)(n2 n) (12...n)^1=(n1 n)
(12...n)^n1 (n1 n)(12...n)^n1=(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 http://www.mathhelpforum.com/mathhe...94de62bf1.gif such that http://www.mathhelpforum.com/mathhe...883c0ba41.gif,
Above I proved that (12)(1...n) generates Sn so therefore http://www.mathhelpforum.com/mathhe...33c6fce11.gif
How do I prove that http://www.mathhelpforum.com/mathhe...6fc81d6c1.gif. so I can establish that G=Sn??
AHHHH

Quote:
Originally Posted by
RoboMyster5 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)(n2 n) (12...n)^1=(n1 n)
(12...n)^n1 (n1 n)(12...n)^n1=(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 http://www.mathhelpforum.com/mathhe...94de62bf1.gif such that http://www.mathhelpforum.com/mathhe...883c0ba41.gif, Above I proved that (12)(1...n) generates Sn so therefore http://www.mathhelpforum.com/mathhe...33c6fce11.gif How do I prove that http://www.mathhelpforum.com/mathhe...6fc81d6c1.gif. so I can establish that G=Sn?? AHHHH
From my understanding, you used the theorem that S_n is generated by n1 transpositions (1,2), (2,3), ..., (n1, n). You showed that (1,2) and (1,2, ... , n) indeed generate those n1 transpostions, which shows that $\displaystyle S_n \subseteq G$. We see that G (a set generated by (1,2) and (1,2,...,n) ) is just the set of permutations of {1,2, ..., n}. By the definition of the symmetric group of degree n, it forces $\displaystyle G \subseteq S_n$. Thus G=S_n, a symmetric group of degree n.
EDIT (some additional remarks): G must contain (1,2) and (1,2, ... , n). Since G is the group by hypothesis, G must contain a set generated by (1,2) and (1,2, ... ,n). Since our choice of G should be minimal, if we show that a set generated by (1,2) and (1,2, ... ,n) is the group, the proof is completed.

Wow I think you just made this make complete sense...I applaud you. Thank you.