Schur multiplier of SL₂ over finite commutative rings

2025/10/04 by Behrooz Mirzaii, Mirzaii, Behrooz, Abraham Rojas Vega +1
Mathematics · Computer Science · #Finite Group Theory Research #Rings, Modules, and Algebras #Coding theory and cryptography

paper · pdf · doi:10.48550/arxiv.2510.03946

Abstract

In this article, we investigate the Schur multiplier of the special linear group SL₂(A) over finite commutative local rings A. We prove that the Schur multiplier of these groups is isomorphic to the K-group K₂(A) whenever the residue field A/\mathfrakm_A has odd characteristic and satisfies |A/\mathfrakm_A| ≠ 3,5,9. As an application, we show that if A is either the Galois ring GR(p^l,m) or the quasi-Galois ring A(p^m,n) with residue field of odd characteristic and |A/\mathfrakm_A| ≠ 3,5,9, then the Schur multiplier of SL₂(A) is trivial.

Citations

• On the connections between the low dimensional homology groups of \textrmSL₂ and \textrmPSL₂
• The abelianization of the elementary group of rank two
• The third homology of projective special linear group of degree two
• Finite local principal ideal rings
• Bloch Groups of Rings
• The third homology of stem-extensions and Whitehead's quadratic functor
• Finite groups of units of finite characteristic rings
• A Bloch-Wigner exact sequence over local rings
• The third homology of SL₂ of local rings
• A Bloch-Wigner complex for SL₂

Related

• On the Exponent of the Schur Multiplier
• Classifying p-groups via their multiplier
• A characterization of finite p-groups by their Schur multiplier
• The Schur multiplier of some finite multiplicative Lie algebras
• Symbolic computation of Schur multipliers with an application to the\n groups of order dividing p⁶