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The map splits if and only if $(n,p)$ is either $(2,3)$ or $(3,2)$. As far as I know, this is a theorem of C.-H. Sah. See Theorem 7 in Sah, Chih-Han, Cohomology of split group extensions, J. Algebra 2 …

Let $G_r=\mathrm{GL}_n(\mathbb{Z}/p^r)$. For $x\in G_r$ the centraliser $C_{G_{r}}(x)$ is abelian iff $x$ is regular iff the reduction mod $p$ of $x$ is regular. This is due to G. Hill, Regular elemen …

The case of $\mathrm{GL}_2$ over a finite local principal ideal ring was treated in a paper by Nagornyj in the late 70s under some (unnecessary) restrictions, and Kutzko obtained similar results indep …

A sufficient criterion for irreducibility is given, for example, in Theorem 4.6 in Hill: Semisimple and cuspidal characters of $\mathrm{GL}_n(\mathcal{O})$. Hill's result is more general, and holds fo …

Let $p$ be a prime; $\mathbb{F}_{p}$ is the field with $p$ elements and $\mathbb{F}_{p}[t]$ the ring of polynomials in $t$ over $\mathbb{F}_{p}$. Does $\mathrm{SL}_{n}(\mathbb{Z}/p^{2})$ have the …

As has been noted, the answer is known for $\mathrm{GL}_2$. For $n>2$ there are certain cases where $\rho:=\mathrm{Ind}_{P(\mathbb{Z}/p^r)}^{\mathrm{GL}_n(\mathbb{Z}/p^r)}\pi$ is irreducible (see for …

As far as I know, there is currently no such explicit formula in the literature. In fact, even for $\mathrm{SO}_3(\mathbb{F}_q)$ (i.e., the case $l=1$), I have not seen a neat table of the dimensions …

I would have preferred to not answer my own question, but here it goes. Yes, the two groups have the same number of conjugacy classes and in fact, the groups $\mathrm{SL}_{n}(W_{2}(\mathbb{F}_{q}))$ a …