# Showing the Heisenberg group admits the presentation $\langle A, B, C\mid AC=CA, BC=CB, ABA^{-1}B^{-1}=C\rangle.$

Consider the subset of $$SL_3(\Bbb Z)$$ consisting of matrices of the form

$$\begin{pmatrix} 1 & a & c\\ 0 & 1 & b \\ 0 & 0 & 1 \end{pmatrix}$$

for $$a,b,c\in\Bbb Z$$.

I need to show that it admits the presentation:

$$\langle A, B, C\mid AC=CA, BC=CB, ABA^{-1}B^{-1}=C\rangle.$$

I am not sure what exactly I need to show. I tried to follow the definition given in class, but didn't really understand it.

Definition 1.36: Let $$G$$ be a group and $$S$$ a generating set. Let $$R\subseteq F(S)$$. Denote by $$\pi$$ the morphism $$F(S)\to G$$. We say that $$G$$ admits the presentation $$\langle S\mid R\rangle$$ if $$R$$ normally generates $$\ker \pi$$, that is, if $$\ker\pi$$ is the smallest normal subgroup containing $$R$$.

Remark 1.38: We can also build a group with any presentation we choose: given a set $$S$$ and a set $$R$$ of words in $$S$$, the quotient $$F(S)/\langle\langle R\rangle\rangle$$ obviously admits the presentation $$\langle S\mid R\rangle$$. Note that this is a way to specify a group algebraically, not geometrically.

• Here's the steps. (1) Choose the correct matrices $A,B,C$ of the form given. (2) Prove the three equations given in the presentation. (3) Prove that any equation in the letters $A,B,C$ and their inverses that holds in the group may be derived formally from the three equations given. How far have you gotten? – Lee Mosher Dec 5 '16 at 13:46
• That's the hard part! For example, I'm sure your choice of $A,B,C$ satisfies the equation $A C B C^{-1} A^{-1} B^{-1} = C$; and you can see that this equation is a consequence of the 2nd and 3rd equations in the presentation, by making a substitution. You have to prove that for every one of the infinitely many equations in the symbols $A,B,C,A^{-1},B^{-1},C^{-1}$ that holds in the group, that equation is a consequence of the three equations given, by a sequence of substitutions. – Lee Mosher Dec 5 '16 at 14:35
• The way I would do this is to show, using the relations of the presentation, that every element in the group defined by the presentation can be represented by a word of the form $A^iB^jC^k$ for $i,j,k \in {\mathbb Z}$. Then show that these elements correspond to distinct elements of the Heisenberg group. So we have an isomorphism between the two groups. – Derek Holt Dec 5 '16 at 16:37
• OK So I Set X={A,B,C} and consider F(X) of X There is a homomorphisim phi:F(X)->H Do I need to show now that R={ACA^-1C^-1,BCB^-1C^-1,....} is a normal subgroup of Ker(phi) – matan Dec 5 '16 at 19:02
• Or use that the Heisenberg group is semidirect product of $Z^2$ and $Z$, where the rank 2 subgroup is normal. – Moishe Kohan Dec 5 '16 at 20:31