Let $n ≥ 2$ and define $V =$ { $x , x+1: 1 ≤ x < n$}.
Show $V$ generates $S_n$ and that it is a minimal generating set for $S_n$.
I'm having a hard time with this. But I do have some ideas on how I can prove this:
First I need to show $V$ generate $S_n$. To do this I was thinking about trying to show that this set generates another set $W$, that is known to generate $S_n$. So trying to show $V$ generates $W =$ {$(1,2) , (1,2,...,n)$}, since I already know that $W$ generates $S_n$ from class. I'm a bit stuck on actually showing this, and I was also wondering if there's a better way to maybe prove $V$ generates $S_n$?
Then, I to show $V$ is a minimal generating set for $S_n$, I was thinking about showing that for any fixed $y$ with $1≤y<n$, the set $M =$ $V$\ {$y,y+1 $} does not generate $S_n$. I'm a bit stuck on actually showing this though.
$V=\{(i,i+1), i=1,...,n-1\}$. Then let $n\ge 3$, we have $(1,2,...,n)=(n-1,n)\cdot ...\cdot (2,3)\cdot (1,2)$ (composition from left to right), so $\langle V\rangle\ge \langle (1,2), (1,2,...,n)\rangle$. Hence $\langle V\rangle=S_n$.
For the second part you need to prove that $(i,i+1)$ is not a product of other transpositions from $V$. That is because the sets $A=\{1,...,i\}, B=\{i+1,...,n\}$ are invariant under the other transpositions from $V$ (hence under their products), but not invariant under $(i,i+1)$.