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I know a presentation of the special linear group SL$(2,3)$(Presentation of ${\rm SL}(2,3)$). My question is that-

Can we give a presentation for SL$(n,\mathbb{Z}_p)$ in general or in more general can we give a presentation for SL$(n,\mathbb{F})$,where $\mathbb{F}$ is field.

As J.P. Serre had given presentation for SL$(2,\mathbb{Z})$

$$\mathrm{SL}_2(\mathbb{Z}) = \langle \,S, T \mid S^4 = 1, (ST)^3 = S^2 \,\rangle$$ where, \begin{align} S &= \begin{pmatrix} \phantom{-}0& 1 \\ -1 & 0 \end{pmatrix}, & T &= \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}. \end{align} I wants it to be more general.

MANI
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  • @DietrichBurde sorry sir, now I have edited it. – MANI Nov 21 '19 at 11:38
  • @DietrichBurde actually sir, I have to do some calculation over SL$(n,\mathbb{F})$, thats why I need this. – MANI Nov 21 '19 at 11:52
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    For a group $G$ there are typically many presentations of $G$, so you mean "I know a presentation of ${\rm SL}(2,3)$", not "I know the presentation of ${\rm SL}(2,3)$". The English word "the" implies uniqueness. There are indeed of presentations of ${\rm SL}(n,F)$ for fields $F$ in the literature (Steinberg presentations for example), and there are a number of other specific presentations of ${\rm SL}(n,q)$ for finite fields. I think this question is really too broad and you should perhaps ask something more limited. – Derek Holt Nov 21 '19 at 11:54
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    For $SL(n,p)$ see the article "Presentations of finite simple groups" by Guralnik et al., section $6.1$. – Dietrich Burde Nov 21 '19 at 11:54

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There are presentations of $SL_n(K)$ for a field $K$, in particular for $K=\Bbb F_p$ and also for $SL_n(\Bbb Z)$. In the latter case we obtain this from the theory of arithmetic groups. The result is listed for $n\ge 3$, for example, in $5.6$, page 14 of the article An introduction to arithmetic groups by Christophe Soule: denote by $x_{ij}$ the matrices with diagonal elements $1$ and another entry $1$ at position $(i,j)$ and zero entries otherwise. Then we have $$ SL_n(\Bbb Z)=\langle x_{ij}, 1\le i\neq j\le n\mid [x_{ij},x_{kl}]=1 \text{ for } j\neq k, i\neq l,\; [x_{ij},x_{jk}]=x_{ik} \text{ for }i,j,k \text{ distinct }, (x_{12}x_{21}^{-1}x_{12})^4=1 \rangle $$ We can also generate the group by two elements only, but the relations then become very long. For the references see [6,9,20] in Soule's lecture notes.

Dietrich Burde
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