If $H$ is a subgroup of $S_n$ contains transpositions $(1,2), (1,3), ..., (1,n)$ then $H = S_n$

Question

I have a question concerning permutations. I'm starter in group theory and would appreciate some ideas. I read something about generators but I didn't understand very well how these work with permutations:

Show if $H$ is a subgroup of $S_n$ contains transpositions $(1,2), (1,3), ..., (1,n)$ then $H = S_n$.

Answer 1

To see this, your observation that $(ij)=(1i)(1j)(1i)$ gives us every transposition.

But every element of $S_n$ can be written as a product of transpositions. For instance, $(abcdef)=(af)(ae)(ad)(ac)(ab)$.