Presentation of Heisenberg Group $\mathbb{H}$ over the field $\mathbb{F}$

Question

Let $\mathbb{F}$ denote the finite field. Denote $\mathbb{H}_{\mathbb{F}}=\left\{ \left( {\begin{array}{ccc} 1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \\ \end{array} } \right)\mid a,b,c \in \mathbb{F}\right\}$ (known as Heisenberg Group over the field $\mathbb{F}$). Could you please tell me the presentation of this group or tell me the reference which contatains the presentation.

Answer 1

If the field has order $p^n$, then you will need $2n$ generators $x_1,\ldots,x_n,y_1,\ldots,y_n$, and it is probably easiest if you throw in $n$ central generators $z_1,\ldots,z_n$, where each of $x_1,\ldots,x_n$, $y_1,\ldots,y_n$, and $z_1,\ldots,z_n$ are generators of the additive group of the field, and they all correspond to each other.

For relations, all generators have order $p$, and all of the $z_i$ commute with all of the $x_i$ and all of the $y_i$. Then you have, for each $i$ and each $j$, a relation $[x_i,y_j]=z_i.z_j$, where the right hand side is multiplication in the field, and you will need to calculate $z_i.z_j$ as a word in the $z_k$.

For example, for the field of order 9, using additive generators $1,w$ with $w$ a primitive element satisfying $w^2=1+w$, the $[x,y]$ commutator relation become

$[x_1,y_1]=z_1$, $[x_1,y_2]=[x_2,y_1]=z_2$, $[x_2,y_2]=z_1z_2$.