Abstract
In this paper we compute the integral homology of the Borel subgroup $B$ of the special linear group $SL(2,\mathbb{F}_p), p$ is a prime number. Arcoding to Adem \cite{AJM} these are periodic groups. In order to compute the integral homology of $B,$ we decompose it into $\ell-$ primary parts. We compute the first summand based on Invariant Theory and compute the rest summand based on Lyndon-Hochschild-Serre spectral sequence. We assume that $p$ is an odd prime and larger than $3.$
Highlights
In the theory of algebraic groups, a Borel subgroup of an algebraic group is a maximal Zariski closed and connected solvable algebraic subgroup
We briefly recite some facts about group cohomology and the transfer homomorphism 1–4, which will be used frequently throughout this paper
If H ⊂ G is a subgroup, the inclusion BH → BG induces a map in cohomology resGH : Hn(G, A) → Hn(H, A) called restriction
Summary
In the theory of algebraic groups, a Borel subgroup of an algebraic group is a maximal Zariski closed and connected solvable algebraic subgroup. Let G be a finite group and A be a G−module, we define If H ⊂ G is a subgroup, the inclusion BH → BG induces a map in cohomology resGH : Hn(G, A) → Hn(H, A) called restriction. On the homology of Borel subgroup of SL(2,Fp). If H contains a Sylow p− subgroup of G and is normal in G, resGH : Hn(G, A)(p) ∼= Hn(H, A)G/H .
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