Mixability of finite groups
Gideon Amir, Guy Blachar, Subhajit Ghosh, Uzi Vishne
TL;DR
We introduce mixability for finite groups via random subproducts and define the mixing length $\ell_{\mathrm{mix}}(G)$, linking it to uniform distribution on $G$ and to existing random-walk perspectives. A representation-theoretic framework is developed: a Decomposition Criterion shows $G$ is mixable iff every nontrivial irreducible representation is mixable, which is analyzed via induced representations and central idempotent decompositions in $\mathbb{C}[G]$; this yields necessary conditions and structural reductions. The paper proves mixability for broad families (2-groups, symmetric and alternating groups with explicit constructions, most finite irreducible Coxeter groups, many matrix groups when $q-1$ is a power of $2$, and several 2-transitive sporadic groups like the Mathieu groups and Higman–Sims), and provides concrete mixing-length bounds. It also offers a conjectural complete obstruction: a finite group is mixable precisely when it has no nontrivial odd-order quotient (i.e., is $2'$-simple), supported by low-dimensional representation analysis and structural reductions, and raises questions about the universality of mixing lengths up to a universal constant. Overall, the work develops a cohesive framework tying group mixability to representation theory and transitive actions, enabling systematic verification across large classes of groups with implications for randomization and uniformity in finite groups.
Abstract
Say that a finite group $G$ is mixable if a product of random elements, each chosen independently from two options, can distribute uniformly on $G$. We present conditions and obstructions to mixability. We show that $2$-groups, the symmetric groups, the simple alternating groups, several matrix and sporadic simple groups, and most finite Coxeter groups, are mixable. We also provide bounds on the mixing length of such groups.