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The aim of this paper is to give all quasitriangular structures on a class ofsemisimple Hopf algebras constructed through abelian extensions of$\Bbbk\mathbb{Z}_{2}$ by $\Bbbk^G$ for an abelian group $G$. We first introducethe concept of symmetry of quasitriangular structures of Hopf algebras andobtain some related propositions which can be used to simplify our calculationsof quasitriangular structures. Secondly, we find that quasitriangularstructures of these semisimple Hopf algebras can do division-like operations.Using such operations we transform the problem of solving the quasitriangularstructures into solving general solutions and giving a special solution. Thenwe give all general solutions and get a necessary and sufficient condition forthe existence of a special solution.

Quasitriangular structures on abelian extensions of $\mathbb{Z}_{2}$