On the modular cohomology of a Sylow p-subgroup of SL_2(Z/p^nZ)

Abstract

Abstract This PhD thesis is set in the greater project of the computation of H∗(GL_2(Z/p^n), F_p)), where p is a prime, n > 1 an integer, Z/p^n is the ring of integers modulo p^n and F_p is the field of p elements. The case for n = 1 and any prime p was solved by Aguade in [AGU] in 1980. For n > 1 the problem remains open. In this thesis some new progress towards a solution has been made. We discuss the subgroups of GL_2(Z/p^n) and SL_2(Z/p^n), and in particular focus on the Sylow-p-subgroups Syl_p(n, S) of SL_2(Z/p^n). The research is presented in two parts: In the first part we examine the Sylow-p-subgroups of GL_2(Z/p^n) and SL_2(Z/p^n) for any p and n > 1. The main result for that part is Theorem 3.2.6, which says the following: We associate the Lyndon-Hochschild-Serre spectral sequence to a given group extension of the Sylow-p-subgroups of GL_2(Z/p^n) and SL_2(Z/p^n). Then the E2 pages do not depend on n. The second part focusses on SL_2(Z/3^n) in particular and presents the computation of H∗(Syl_3(n), F_3)) explicitly, with main result Theorem 4.2.4. Finally we conclude with some thoughts about future steps in the underlying project.

Date of Award	6 Jan 2025
Original language	English
Awarding Institution	The University of Manchester
Supervisor	Peter Symonds (Supervisor) & Charles Eaton (Supervisor)