Bounding the order of complex linear groups and permutation groups with selected composition factors

Abstract:

Originally motivated by questions of P. Etingof (related to growth rates of tensor powers in symmetric tensor categories, see K. Coulembier, P. Etingof, and V. Ostrik [Asymptotic properties of tensor powers in symmetric tensor categories, arXiv:2301.09804, 2023]), we obtain (as usual, modulo Abelian normal subgroups) general bounds of exponential nature on the order of finite subgroups $G$ of $\operatorname {GL}(n,\mathbb {C})$ with prescribed properties. Our most general result of this nature is:

\theorem* Let $\mathbf {P}$ be a property of (isomorphism types of) finite groups which is inherited both by subgroups, and by homomorphic images. Then one and only one of the following is true:

1. Every finite group has property $\mathbf {P}$.
2. There is a real number $c(\mathbf {P})$ such that whenever $n$ is a positive integer and $G$ is a finite subgroup of $\operatorname {GL}(n,\mathbb {C})$ which has property $\mathbf {P}$, then $G$ has an Abelian normal subgroup $A$ with $[G:A] \leq c(\mathbf {P})^{n-1}$.

\endtheorem*

We prove this theorem as a consequence of a more precise result on properties $\mathbf {Q}$ which are inherited by normal subgroups and homomorphic images, where (using the Classification of the Finite Simple Groups), we are able to determine exactly when such a constant $c(\mathbf {Q})$ exists, and to give explicit upper bounds for it when it does. Note that since the symmetric group $\mathrm {Sym}_{n+1}$ is isomorphic to a subgroup of $\operatorname {GL}(n,\mathbb {C})$ for all $n$, the two options in the theorem are indeed mutually exclusive.

The two questions from P. Etingof concerned groups of order prime to $p$, and groups with Abelian Sylow $p$-subgroups, respectively, where $p$ is a specified prime. We are able to provide the optimal upper bound in almost every case. By placing the questions in this general framework, we are able to indicate numerous other directions to apply our methods.