Abstract This PhD thesis is set in the greater project of the computation of Hâ(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 Sylowpsubgroups Syl_p(n, S) of SL_2(Z/p^n). The research is presented in two parts: In the first part we examine the Sylowpsubgroups 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 LyndonHochschildSerre spectral sequence to a given group extension of the Sylowpsubgroups 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 Hâ(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  31 Dec 2024 

Original language  English 

Awarding Institution   The University of Manchester


Supervisor  Peter Symonds (Supervisor) & Charles Eaton (Supervisor) 

On the modular cohomology of a Sylow psubgroup of SL_2(Z/p^nZ)
Meyer, A. (Author). 31 Dec 2024
Student thesis: Phd