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On the homology of Borel subgroup of SL(2,Fp)

In this paper we compute the integral homology of the Borel subgroup B of the special linear group SL(2,Fp) where p is a prime. In order to compute the integral homology of B, we decompose it into ℓ− primary parts. | Science Technology Development Journal 22 3 308-313 Research Article On the homology of Borel subgroup of SL 2 Fp Bui Anh Tuan Vo Quoc Bao ABSTRACT In the theory of algebraic groups a Borel subgroup of an algebraic group is a maximal Zariski closed and connected solvable algebraic subgroup. In the case of the special linear group SL2 over finite fields Fp the subgroup of invertible upper triangular matrices B is a Borel subgroup. According to Adem1 these are periodic groups. In this paper we compute the integral homology of the Borel subgroup B of the special linear group SL 2 Fp where p is a prime. In order to compute the integral homology of B we decompose it into I primary parts. We compute the first summand based on Invariant Theory and compute the rest based on Lyndon-Hochschild-Serre spectral sequence. In conclusion we found the presentation of B and its period. Furthermore we also explicitly compute the integral homology of B. Keywords Ring cohomology of p-groups periodic groups Invariant Theory Lyndon-Hochschild-Serre spectral sequence Falcuty of Mathematics and Computer Science Ho Chi Minh University of Science Vietnam Correspondence VoQuoc Bao Falcuty of Mathematics and Computer Science Ho Chi Minh University of Science Vietnam Email voquocbao0603@gmail.com History Received 2018-12-04 Accepted 2019-03-22 . Published 2019-08-19 DOI https doi.org 10.32508 stdj.v22i3.1225 PRELIMINARIES For reference we briefly recite some facts about group cohomology and the transfer homomorphism1-4 which will be used frequently throughout this paper. Let G be a finite group and A be a G module then we define Hn G A Hn BG A where Bg is classifying space of the group G. The group Hn G A is called the cohomology group of G with untwisted coefficient A. If H c G is a subgroup the inclusion Bh Bg induces a map in cohomology resG Hn G A Hn H A called restriction. Because inner automorphisms of G act trivially on cohomology we have Im resG is contained in Hn H A Ng h h. There is .