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 $$ \langle t, c \,|\,t^2, c^n, (tc)^{n-1}, [t, c]^3, [t, c^k]^2 = 1 \text{ for } 2 \leq k \leq n/2\rangle $$

is indeed the presentation you need. The best (to my knowledge) way to check its validity is first writing down Coxeter presentation as a reflection group, and then eliminating extra generators replacing them by products of conjugates of a transposition.

  • $\begingroup$ Where square brackets denotes the commutator? $\endgroup$ – Matt Jan 4 at 3:46
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    $\begingroup$ 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. :) $\endgroup$ – xsnl Jan 4 at 3:53
  • $\begingroup$ Thank you!!! :) $\endgroup$ – Matt Jan 4 at 11:06
  • $\begingroup$ I don't believe this is the correct presentation for all $n$ (doesn't work for $n=6$ for example). $\endgroup$ – Steve D Jan 4 at 19:02
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    $\begingroup$ Oh I see, you have a typo: should be $[t,c^k]^2$ in the last relation. $\endgroup$ – Steve D Jan 4 at 19:13

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