# What are the relations in this presentation of $S_n$?

The permutation group $$S_n$$ can be generated by an element $$(i \; \; i+1)$$ together with an $$n$$-cycle.

First, I would like to check that the above statement is indeed true?

My main question is, is it known what the relations are when we generate $$S_n$$ like this?

Well, let's say that first one is on you. For the other one, one can (somewhat easily) find out that $$\langle t, c \,|\,t^2, c^n, (tc)^{n-1}, [t, c]^3, [t, c^k]^2 = 1 \text{ for } 2 \leq k \leq n/2\rangle$$
• Yes. A forewarning for a person diving into group theory literature: always note if the author right or left, i. e. uses $[a, b] = a^{-1}b^{-1}ab$ or $[a, b] = aba^{-1}b^{-1}$. It does not matter in my answer, but reading two texts on same subject with different chirality can get quite annoying. :) – xsnl Jan 4 at 3:53
• I don't believe this is the correct presentation for all $n$ (doesn't work for $n=6$ for example). – Steve D Jan 4 at 19:02
• Oh I see, you have a typo: should be $[t,c^k]^2$ in the last relation. – Steve D Jan 4 at 19:13