Let $p$ be an odd prime, and consider the group
$$\{U\in \operatorname{SL}\_n(\mathbb{Z}/p\mathbb{Z}) : U^{t}U=I \bmod p \}\subseteq \operatorname{SL}\_n(\mathbb{Z}/p\mathbb{Z}).$$


I wonder what is the structure of this subgroup? Can the generators of this subgroup be written or calculated ?


We know that if $U^{t}U=I$ with $U\in \operatorname{GL}\_n(\mathbb{Z})$, then $U$ must be a signed permutation. So I came up with the above subgroup.



A subgroup of $\mathrm{SL}_n(\mathbb{Z}/p\mathbb{Z})$
This is the [special orthogonal group](https://groupprops.subwiki.org/w/index.php?title=Orthogonal_group_for_the_standard_dot_product) over the field $\mathbb{Z}/p \mathbb{Z}$. Or, rather, one of the "special orthogonal groups". For any invertible symmetric matrix $Q$, one can consider the group of matrices $U$ in $\text{SL}\_n$ such that $U Q U^T = Q$, which forms a subgroup $\text{SO}(Q)$. If one allows matrices in $\text{GL}\_n$ instead, this is $O(Q)$.


I'll limit myself to $p$ odd throughout.


**"Split" versus "non-split"** If $n=2m+1$ is odd, then any two nondegenerate quadratic forms define the same orthogonal group.


If $n=2m$ is even, then there are two classes of quadratic forms, called the "[split](https://groupprops.subwiki.org/wiki/Split_orthogonal_group)" and the "non-split" class. The case you have described is split if $p^{m} \equiv 1 \bmod 4$ and is non-split if $p^{m} \equiv 3 \bmod 4$. I'll denote the split and non-split cases by $SO\_{2m}^+(p)$ and $SO\_{2m}^-(p)$.


**Order of the group** The order of $SO\_{2m+1}$ is
$$p^{m^2} \prod\_{i=1}^m (p^{2i}-1).$$
The orders of $SO\_{2m}^+(p)$ and $SO\_{2m}^-(p)$ are
$$p^{m(m-1)} (p^m-1) \prod\_{i=1}^{m-1} (p^{2i}-1) \ \text{and} \ p^{m(m-1)} (p^m+1) \prod\_{i=1}^{m-1} (p^{2i}-1)$$
respectively. Note, in particular, that they can't be isomorphic since they have different sizes.


**Generators** The orthogonal group $O\_n$ is generated by reflections; for a vector $x$ with $x \cdot x \neq 0$, the reflection over $x$ is the linear map $r\_x(v) := v - \tfrac{x \cdot v}{x \cdot x} x$. Reflections have determinant $-1$, so the special orthogonal group $SO\_n$ is generated by pairs of reflections; the product $r\_x r\_y$ fixes the $(n-2)$-plane $x^{\perp} \cap y^{\perp}$, so we can also say that $SO\_n$ is generated by rotations fixing planes of dimension $n-2$. A shorter list of generators can be found in [Matrix Generators for the Orthogonal Groups](https://www.maths.usyd.edu.au/u/ResearchReports/Algebra/RylTay/1997-7.html), Rylands and Taylor , 1998.


**Structure** The orthogonal group $O\_n$ has two characters to $\{ \pm 1 \}$: The determinant map, which sends every reflection to $-1$, and the [spinor norm](https://groupprops.subwiki.org/wiki/Spinor_norm), which sends the reflection $r\_x$ to $\left( \tfrac{x \cdot x}{p} \right)$ (this is the quadratic residue symbol). I'll write $K\_{2m+1}$, $K\_{2m}^+$ or $K\_{2m}^-$ for the common kernel of these characters. So $K$ is an indexed $2$ subgroup of the corresponding $SO$.


For $n=2m+1$ odd, the group $K\_n$ is the [Chevalley group](https://groupprops.subwiki.org/wiki/Chevalley_group_of_type_B) $B\_m(p)$. The group $B\_1(p)$ is isomorphic to $\text{PSL}\_2(p)$, which is the alternating group $A\_4$ if $p =3$ and is otherwise simple. I believe that $B\_m(p)$ is simple for all $m \geq 2$ and $p \geq 3$, but I couldn't find a reference for this.


For $n$ even in the case you care about, where $Q = \text{Id}\_n$, the group $K\_n$ contains $- \text{Id}\_n$, which generates the center. For the other quadratic form that you didn't use, the spinor norm of $- \text{Id}\_n$ is $-1$, and the center of $K\_{n}$ is trivial.


The quotients of $K\_{2m}^+(p)$ and $K\_{2m}^-(p)$ by their centers are the [Chevalley group](https://groupprops.subwiki.org/wiki/Chevalley_group_of_type_D) $D\_m(p)$ and the twisted Chevalley group ${}^2 D\_m(p^2)$ respectively. We have $D\_2(p) \cong \text{PSL}\_2(p) \times \text{PSL}\_2(p)$. Again, $\text{PSL}\_2(p)$ is simple for any $p \geq 5$, whereas $\text{PSL}\_2(3) \cong A\_4$. I believe that $D\_m(p)$ and ${}^2 D\_m(p^2)$ are simple for $m \geq 3$ (and $p$ odd), but again I couldn't find a reference


I'm using [Wikipedia](https://en.wikipedia.org/wiki/List_of_finite_simple_groups#Groups_of_Lie_type) as my source for group theoretic notation.


