Presentation of $S_4$ with $3$ generators and easier relators

Question

The presentation $$< x^4,y^3,z^2,yxzx^2,zxzy^2x^3,zyzy >$$ for $S_4$ is quite complicated. $S_4$ could be easily generated by $2$ elements, but I prefer $3$ generators with orders $2,3$ and $4$ because then the elements $x^j\cdot y^k\cdot z^l$ with $0\le j\le 3\ ,\ 0\le k\le 2\ ,\ 0\le l\le 1$ are the group elements.

This is impossible with $2$ generators because the maximum order of an element in $S_4$ is $4$.

In general, $n-1$ generators are sufficient for $S_n$ , if the above property is required (maybe, less relators suffice for larger $n$, does anyone know the actual minimum ? )

Is there a presentation of $S_4$ with $3$ generators and easier relators ? The relators must determine $S_4$ uniquely. It would be best, if the presentation could be generalized for $S_n$ and $n-1$ generators.

Answer 1

$\langle x,y,z \mid x^2=y^3=z^4=xyz=1 \rangle$ is a nice presentation of $S_4$ on three generators. I don't know anything equivalent for $S_n$ - probably the Coxeter presentation described by Matt Samuel is the "nicest".

Many people were surprised when it was proved that there exist are presentations of $S_n$ on two generators and a fixed number of relations. See, for example here, where such a presentation is described with at most $123$ relations, for all $n$. You woudl not describe these as nice presentations, but they are useful in computational applications.

It has now been proved that all finite simple groups, with the possible exception of the Ree groups $^2G_2(3^{2k+1})$ have presentations on two geenrators and a bounded number of relators.

Answer 2

You could use the Coxeter generators, $s_1=(12)$, $s_2=(23)$, $s_3=(34)$. Then the relations are $$s_1^2=s_2^2=s_3^2=(s_1s_2)^3=(s_2s_3)^3=(s_1s_3)^2=1$$ This generalizes similarly to $S_n$, $$s_i^2=1$$ $$(s_is_{i+1})^3=1$$ $$(s_is_j)^2=1$$ if $|i-j|>1$.

This can be made to satisfy the element order condition by using the generators $s_1s_2\cdots s_k$ instead, which are the cycles $(12\ldots k)$. Modify the relations accordingly.