# Presentation of a group with the same number of generators and relations

This may be a silly question as I do not know much about presentations of groups.

Let $$\langle S\mid R\rangle$$ be a finite presentation of a group. Is it always possible to get a finite presentation $$\langle S’\mid R’\rangle$$ of an isomorphic group with $$|S’|=|R’|$$, at least if $$|R|>0$$?

I know that if $$|S|>|R|$$ I can simply double some relations until I get the equality, but it is not clear to me if I can do something analogous when $$|S|<|R|$$, since when adding a new generator I must also add a new relation.

• What is S and what is R Sep 14, 2021 at 15:02
• A group is a set plus a binary operation denoted as (Set,Binary operation) .. I don't understand your notation.. or are you talking about a group in a sense other than abstract algebra Sep 14, 2021 at 15:02
• You can always put infinitely many generators and relations. Then $|R|=|S|=\infty$. Sep 14, 2021 at 15:06
• @Buraian This is standard notation for a group presentation. Sep 14, 2021 at 15:22
• I should have clarified the notation. I also added the condition that $R$ and $S$ be finite. Sep 14, 2021 at 15:24

The answer is no. The smallest example is the presentation of the Klein $$4$$-group $$\langle x,y \mid x^2 = y^2 = 1, xy=yx\rangle.$$ In any presentation $$\langle S \mid R \rangle$$ of this group we have $$|R| \ge |S| + 1$$.
I don't know whether there is an elementary proof of that. It follows from a more general result that, for a finite presentation $$\langle S \mid R \rangle$$ of a finite group $$G$$, we have $$|R| \ge |S| + |d(M(G))|$$, where $$d(M(G))$$ is the smallest number of generators of the Schur Multiplier $$M(G)$$ of $$G$$.
$$M(G)$$ is the (unique up to isomorphism) largest group $$M$$ for which there is a group $$E$$ with $$M \le Z(E) \cap [E,E]$$ and $$E/M \cong G$$. For the Klein 4-group, we have $$|M(G)| = 2$$, where, for the covering group $$E$$ we can take either $$D_8$$ or $$Q_8$$.
Interestingly, $$Q_8$$ does have a balanced presentation (i.e. $$|S|=|R|$$), namely $$\langle x,y \mid x^2=y^2, y^{-1}xy=x^3\rangle$$.