Rota-Baxter operators on dihedral and alternating groups

Abstract

Rota-Baxter operators on algebras, which appeared in 1960, have connections with different versions of the Yang-Baxter equation, pre- and postalgebras, double Poisson algebras, etc. In 2020, the notion of Rota-Baxter operator on a group was defined by L. Guo, H. Lang, Yu. Sheng. In 2023, V. Bardakov and the second author showed that all Rota-Baxter operators on simple sporadic groups are splitting, i. e. they are defined via exact factorizations. In the current work, we clarify for which $n$, there exist non-splitting Rota-Baxter operators on the alternating group $\mathrm{A}_n$. For the corresponding $n$, we describe all non-splitting Rota-Baxter operators on $\mathrm{A}_n$. Moreover, we describe Rota-Baxter operators on dihedral groups $D_{2n}$ providing the general construction which lies behind all non-splitting Rota-Baxter operators on $\mathrm{A}_n$ and $D_{2n}$.