# How quasirandom are the nonabelian finite simple groups?

A group is $$d$$-quasirandom if every nontrivial complex representation has dimension at least $$d$$. Gowers introduced quasirandomness in this paper and proved that every nonabelian finite simple group of order $$n$$ is $$\sqrt{\log n}/2$$-quasirandom.

Question: What is the correct (asymptotic) lower bound for the quasirandomness of nonabelian finite simple groups?

Gowers indicates that a theorem of Jordan (Theorem 14.12 here) implies a slightly better bound. Indeed, this theorem gives that a nonabelian simple group of order $$n$$ has a nontrivial representation of dimension $$d$$ only if $$n\leq (d!)12^{d(\pi(d+1)+1)}$$, which implies that these groups are $$\Omega(\sqrt{\log n\log\log n})$$-quasirandom.

Of course, the correct bound should be obtainable using the classification of finite simple groups. The sporadic groups are irrelevant to the asymptotics, and the alternating groups are $$\Omega(\log n/\log\log n)$$-quasirandom. It remains to study the groups of Lie type. What I want can be read off the first column of the generic character tables of these groups, but to my surprise, these aren't known yet (see LeechLattice's answer here). Interestingly, the answer here describes how to find an upper bound on the smallest nontrivial representation in these cases, whereas I want a lower bound.

• A natural way of writing this is to ask for the largest finite simple subgroup of the unitary group $U(n)$, and the obvious guess is $A_{n+1}$. Dec 2 '21 at 11:56
• For finite simple groups, rather than all finite groups, I think the minimal non-trivial projective dimension (i.e., dimension of a minimal representation of a central extension) is known explicitly. (There might be a couple of cases where it's not completely known down to the exact number, but I think for the complexes it is.) The papers you are looking for are the unique ones by Landazuri and Seitz, and a follow up by Seitz and Zalesski, There have been a few others since then, but their bounds are usually pretty good. Dec 3 '21 at 9:09

Suppose $$G$$ is a finite simple group of order $$n$$ with a nontrivial representation of degree $$d$$. Then $$G$$ is isomorphic to a subgroup of $$U(d)$$. By Collins's sharp version of Jordan's theorem (https://www.degruyter.com/document/doi/10.1515/JGT.2007.032/html), $$G$$ has an abelian normal subgroup of index at most $$(d+1)!$$, which must be trivial since $$G$$ is simple, so $$|G| \leq (d+1)!$$. Rearranging, $$d \gtrsim \log n / \log\log n$$.
Edit: Specializing to simple $$G$$ actually reduces Collins's paper to a reference to a reference to the paper of Seitz and Zalesskii (https://www.sciencedirect.com/science/article/pii/S0021869383711324?via%3Dihub) mentioned by David Craven in the comments, so that's really the heart of the matter. We thereby get the slightly stronger bound $$n \leq (d+1)!/2$$ (for sufficiently large $$d$$). Apart from alternating groups I think you can read out a much stronger bound like $$n \leq \exp O((\log d)^2)$$, or $$d \geq \exp \Omega( (\log n)^{1/2})$$ (I am guessing the next worst case is $$\mathrm{SL}_n(2)$$).