Quasirandom and quasisimple groups

Quasirandom and quasisimple groups, Discrete Analysis 2025:21, 24 pp.

A quasirandom family of groups is a sequence of finite groups $(G_n)$ such that the smallest dimension of a non-trivial irreducible representation of $G_n$ tends to infinity with $n$. The terminology “quasirandom group” was introduced by Gowers because quasirandom groups have properties that are closely related to properties of quasirandom graphs, but the concept is implicit in earlier work of Sarnak and Xue.

Gowers used quasirandom groups to answer a question of Babai and Sós, by showing that there are groups $G$ of order $n$ such that the largest product-free subset of $G$ (that is, a subset that contains no triple $x,y,z$ with $xy=z$) has size $o(n)$. The upper bound he obtained was roughly $n/d^{1/3}$, where $d$ is the dimension of the smallest non-trivial representation of $G$. For this to answer Babai and Sós’s question, one must exhibit a quasirandom family of groups, but such families were well known to exist: in fact, most natural families of non-Abelian groups, in particular all families of simple groups, are quasirandom.

However, among these families, one observes strikingly different quantitative behaviour. Typically, groups of Lie type, such as $\mathrm{PSL}_2(q)$, are “highly quasirandom” in the sense that their smallest irreducible representations have a dimension that is a power of $|G|$, whereas the alternating group $A_n$ with $n\geq 7$, for example, has smallest non-trivial irreducible representation of dimension $n-1$, which is logarithmic in $|G|$. It is therefore interesting to try to understand what it tells us about a group if its smallest non-trivial representations are not just large but very large.

To that end, this paper defines a group $G$ to be $\epsilon$-quasirandom if its smallest non-trivial irreducible representation has dimension at least $|G|^\epsilon$, and it provides a description of $\epsilon$-quasirandom groups.

Recall that the solvable radical of a group $G$ is the largest solvable normal subgroup of $G$. One of the main results of this paper is that if $G$ is $\epsilon$-quasirandom, then the quotient of $G$ by its solvable radical has size at least $|G|^\epsilon$ and is a direct product of boundedly many finite simple groups of Lie type of bounded rank. In the other direction, if $G$ is perfect (that is, has no non-trivial Abelian quotient), and the quotient of $G$ by its solvable radical has size at least $|G|^\epsilon$ and is $\epsilon$-quasirandom (which is in particular the case if it is a product of sufficiently few finite simple groups of Lie type with sufficiently small rank), then $G$ is $\epsilon^2/3$-quasirandom. Thus, loosely speaking a highly quasirandom group $G$ is one with a solvable radical of size bounded above by a non-trivial power of $|G|$ such that the quotient is a product of boundedly many simple groups of Lie type and bounded rank.

The other main aim of the paper is to characterize $\epsilon$-quasirandom groups more precisely when $\epsilon$ is not just bounded away from zero but some fixed constant. It turns out that $\epsilon=1/5$ is an interesting threshold. The authors show that with the exception of one group and one family of groups that they identify, all $\frac 15$-quasirandom groups are quasisimple: a group $G$ is said to be quasisimple if it is perfect, and the quotient by the centre is a non-Abelian finite simple group. Quasisimple groups can be thought of as slightly thickened simple groups, so this result is coming close to saying that $\frac 15$-quasirandom groups are finite simple groups. In any case, it gives a complete description.

The exceptional family is $\epsilon$-quasirandom for every $\epsilon<3/14$, so a corollary of this second theorem is that all $\frac 3{14}$-quasirandom groups are quasisimple.