2024/01/16 by Carl‐Fredrik Nyberg‐Brodda, Nyberg-Brodda, Carl-Fredrik
Mathematics · #13D03 (primary) #20F05 (secondary) #20H25 #Advanced Algebra and Geometry #Algebraic structures and combinatorial models #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #K-Theory and Homology (math.KT)
paper · pdf · doi:10.48550/arxiv.2401.08146
openalex publication_date 2024/01/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For all m ≥ 1, we prove that the abelianization of SL2(ℤ[(1)/(m)]) is (1) trivial if 6 | m; (2) ℤ / 3ℤ if 2 | m and gcd(3,m)=1; (3) ℤ / 4 ℤ if 3 | m and gcd(2,m)=1; and (4) ℤ / 12ℤ ≅ ℤ / 3ℤ × ℤ / 4ℤ if gcd(6,m)=1. This completes known computational results of Bui Anh & Ellis for m ≤ 50. The proof is completely elementary, and in particular does not use the congruence subgroup property. We also find a new presentation for SL2(ℤ[(1)/(2)]). This presentation has two generators and three relators. Thus, SL2(ℤ[(1)/(2)]) admits a presentation with deficiency equal to the rank of its Schur multiplier. This also gives new and very simple presentations for the finite groups SL2(ℤ / m ℤ), where m is odd.