# Minimal number of relations for finite $p$-groups

From the (sharpened) Golod/Shafarevich inequality we know that for finite $p-$groups, where $r$ is the minimal number of relations and $d$ is the minimal number of generators, that $r > \frac{d^2}{4}$

For the symmetric group of degree $3$, $S_3$, I know that $d = 2$ and $r = 2$. But can someone explain to me what two generators we can use and demonstrate that these generators and relations (defined abstractly without already using the permutations) give us the multiplication table for $S_3$? Also, back to $p-$groups, the above inequality for $d = 4$ would give us that $r\ge 5$.

Can someone give me an example of a $2-$group with $d = 4$ and $r = 5$?

• What's a presentation of $S_3$ with $r=2$? (specifically, how do you know that $r=2$?) – Steven Stadnicki Dec 13 '14 at 5:20

## 1 Answer

$$\langle a,b \mid a^2=1, a^{-1}ba=b^2 \rangle$$

is a presentation of $S_3$.

Note that the relations imply $$b = a^{-2}ba^2=a^{-1}(a^{-1}ba)a = a^{-1}b^2a = (a^{-1}ba)^2 = b^4$$

giving $b^3=1$, and now it is easy to see that $G \cong S_3$.

Added later: More generally, for any $k \ge 1$, $$\langle a,b \mid a^2=1, a^{-1}b^ka = b^{k+1} \rangle$$ is a presentation of the dihedral group of order $2(2k+1)$. The proof is left to the reader.

• Thank you very much, Derek. I don't think I would have thought of writing "b" that way to begin with. And any thoughts about the second part of my question, pertaining to a 2-group (or even any p-group) with d = 4 and r = 5? The reason I'm interested in this is that I am actually hoping that for 2-groups it is always the case that r is greater than or equal to 6 when d = 4, which would establish in class field theory the infinite 2-class field tower conjecture for imaginary quadratic number fields with 2-class group of rank 4, conjectured in the 1970s. – Elliot Benjamin Dec 15 '14 at 5:12