Schur multiplier and Schur covers of relative Rota-Baxter groups

2023/11/21 by Pragya Belwal, Belwal, Pragya, Nishant Rathee +3
Mathematics · #Advanced Topics in Algebra #Finite Group Theory Research #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.2311.12384

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 M_RRB(A) of a relative Rota-Baxter group A =(A,B,β,T), and prove that the exponent of M_RRB(A) divides |A||B| when A is 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.

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