Schur multiplier and Schur covers of relative Rota–Baxter groups

Abstract

Relative Rota–Baxter groups are generalizations of Rota–Baxter groups and share a close connection with skew left braces. These structures are well-known for offering bijective non-degenerate set-theoretical solutions to the Yang–Baxter equation. This paper builds upon the recently introduced extension theory and low-dimensional cohomology of relative Rota–Baxter groups. We prove an analogue of the Hochschild–Serre exact sequence for central extensions of relative Rota–Baxter groups. We introduce the Schur multiplier MRRB(A) of a relative Rota–Baxter group A=(A,B,β,T), and prove that the exponent of MRRB(A) divides |A||B| when both A and B are finite. We define weak isoclinism of relative Rota–Baxter groups, introduce their Schur covers, and prove that any two Schur covers of a finite bijective relative Rota–Baxter group are weakly isoclinic. The results align with recent results of Letourmy and Vendramin for skew left braces.