Question. Let $G$ be a finite group. Can we find some (commutative) ring $R$ and some positive integer $d=O(\log\lvert G\rvert)$ such that $G$ can be found as a subgroup of $\operatorname{PGL}_d(R)$?
Motivation. There are a handful of results of the form "every finite group is a subgroup of some group of 'X type'." For example, Cayley's theorem says that every finite group is a subgroup of some symmetric group $S_d$. This also implies that every finite group is a subgroup of $\operatorname{GL}_d(\mathbb F)$ for some $n$ and any field $\mathbb F$; the study of such representations forms, well, representation theory.
One may hope for something stronger: namely, one may hope for decent quantitative control on $d$ in the above in terms of the order $N=\lvert G\rvert$. In Cayley's theorem, we have $d=N$, and this can't generally be improved (e.g. take $G=\mathbb Z/N\mathbb Z$). One encounters a similar issue for representation theory over a fixed base field. In characteristic zero, one example is the affine group $G:=\operatorname{Aff}(\mathbb F_q)$, which only has faithful irreducible representations in dimensions at least $p-1\sim N^{1/2}$. However, $\operatorname{Aff}(\mathbb F_p)$ has, by definition, a faithful $2$-dimensional representation over $\mathbb F_q$. Similarly, some simple groups like $\operatorname{PSL}_2(\mathbb F_q)$ fail even to have nontrivial small-dimensional representations, but they do have small-dimensional faithful projective representations over $\mathbb F_q$, arising directly from the definition.
So, perhaps the correct "universal objects" are $\operatorname{PGL}_d(\mathbb F)$ for various fields $\mathbb F$. This is enough to cover every finite simple group:
- a cyclic group $\mathbb Z/p\mathbb Z$ has a faithful one-dimensional representation, and thus a two-dimensional projective representation, over $\mathbb C$;
- the alternating groups $A_d$ have $d$-dimensional projective representations over any field, and $d\sim\frac{\log N}{\log\log N}$;
- the classical Chevalley groups are defined as subgroups of $\operatorname{PGL}_d(\mathbb F)$ for some finite field $\mathbb F$ and $d\lesssim\sqrt{\log N}$;
- the exceptional groups of Lie type are (I think) defined as subgroups of $\operatorname{PGL}_d(\mathbb F)$ for some finite field $\mathbb F$ and $d=O(1)$;
- the sporadic groups have representations over any field of dimension $O(1)$.
I think, however, if one takes a direct product of Lie-type groups defined over different characteristics, such as $\operatorname{PSL}_d(\mathbb F_{p_1})\times\operatorname{PSL}_d(\mathbb F_{p_2})$ for primes $p_1\neq p_2$, one can no longer find a small-dimensional projective representation over a single field. However, the isomorphism $$\operatorname{GL}_d(\mathbb Z/p_1p_2\mathbb Z)\cong\operatorname{GL}_d(\mathbb F_{p_1})\times\operatorname{GL}_d(\mathbb F_{p_2})$$ gives us a $d$-dimensional projective representation $\operatorname{PSL}_d(\mathbb F_{p_1})\times\operatorname{PSL}_d(\mathbb F_{p_2})$ over the ring $\mathbb Z/p_1p_2\mathbb Z$. I'm not aware of any additional obstructions, hence the question.
A couple of notes:
If $G$ is abelian, then $G$ is the additive group of some commutative ring $R$, and so $r\mapsto\begin{pmatrix}1&r\\&1\end{pmatrix}$ gives an injection $G\hookrightarrow\operatorname{PGL}_2(R)$.
If one only wants the map $G\to\operatorname{PGL}_d(R)$ to be nontrivial (instead of injective), the above argument for simple groups is enough to give the result, since every group $G$ has a simple quotient.
I don't know of any argument showing that one can't beat $d=O(\log\lvert G\rvert)$ for this phrasing of the question. However, I believe (but haven't yet shown) that the symmetric group $S_n$ is not a subgroup of $\operatorname{PGL}_{n-2}(R)$ for large $n$ and any ring $R$. (Edit: this should probably be $n-3$; see comments of Dave Benson.) This would indicate that one can't beat $\frac{\log\lvert G\rvert}{\log\log\lvert G\rvert}$.
