A short proof of a bound on the size of finite irreducible semigroups of rational matrices

TL;DR

The paper addresses the problem of bounding the size of finite irreducible semigroups of $n\times n$ rational matrices by presenting a concise, representation-theoretic proof that achieves $|S|\le 3^{n^2}$. The method leverages Rhodes's generalized group mapping structure and a Minkowski-style modular reduction, embedding $S$ into a finite linear group modulo a prime and applying a torsion-free (or 2-torsion) kernel property to obtain injectivity. It shows that the bound follows from an injective projection modulo $p$ (with $p=3$ in the general case) and clarifies the special case when the maximal subgroup has odd order, yielding a tighter bound. The result connects finite semigroup theory with classical number-theoretic techniques to provide a simpler route to the bound and deepens understanding of irreducible semigroups of rational matrices.

Abstract

I give a short proof of a recent result due to Kiefer and Ryzhikov showing that a finite irreducible semigroup of $n\times n$ matrices has cardinality at most $3^{n^2}$.

Theorems & Definitions (10)

• Lemma 1: Rhodes Rhodeschar
• proof
• Proposition 2
• proof
• Lemma 3
• proof
• Theorem 4: Minkowski
• proof
• Theorem 5
• proof