Abstract Let p be a an odd prime and let G finite -group with cyclic commutator subgroup $G^{\prime }$ G ′ . We prove that the exponent abelianization of centralizer in are determined by group algebra over any field characteristic If, additionally...

Associated to a convex integral polygon $N$ is cluster integrable system $\mathcal X\_N$ constructed from the dimer model. We compute group $G\_N$ of symmetries X\_N$, called (2-2) modular group, showing that it certain abelian conjectured by Fock and Marshakov. Combinatorially, non-torsion elements are ways shuffling underlying bipartite graph, generalizing domino-shuffling. Algebro-geometrica...