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The Gromov-Lmvson-Rosenberg conjecture for a group G states that a compact spin manifold with fundamental group G admits a metric of positive scalar curvature if and only if a certain topological obstruct ion vanishes. It is known to be true for G = 1 . if 0 has periodic cohomology, and if 0 is a free group, free abelian group, or the fundamental group of an orient able surface. It is also known to be false for a large class of infinite groups. However, there are no known counterexamples for finite groups. In this dissertation we will give a general outline of the positive scalar curvature problem, and sketch proofs of some of the known positive and negative results. We will then focus on finite groups, and proceed to prove the conjecture for the Klein 4-group, all dihedral groups, the semi-dihedral group of order 16, and the rank three group (Z2)3. Throughout the thesis. ko will represent the connective real K-theory spectrum, and KO the periodic real K-theory, and Z2 the field with two elements. It turns out that that the topological obstruction in question lies in the connective real homology ko*(BG) of the classifying space of G. Indeed, our method of proof is to first sketch calculations of ko* (BG), using the techniques and calculations of Bruner and Greenlees. We then give explicit geometric constructions to produce sufficiently many manifolds of positive scalar curvature.

Our aim is to transfer many of the foundational results from the modular representation theory of finite groups to the wider context of profinite groups. We are thus interested in profinite modules over the completed group algebra k[[G]] of a profinite group G, where kc is a finite field of characteristic p. Our approach is as follows. We define the concept of relative projectivity for a profinite module over k[[G]] and prove a characterization analogous to the finite case with additions of interest to the pro and sources for indecomposable finitely generated k[[G]]-modules, extending several results known to hold in the finite case. For sources this requires additional assumptions. We prove a direct analogue of Green's Indecomposability Theorem for finitely generated modules over a virtually pro-p group, as well as a lesser known variant due to M.E. Harris. We give a version of the Green Correspondence for finitely generated modules over virtually pro-p groups.

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Let $V$ be a non-zero finite dimensional vector space over the finite field $F_q$. We take the left action of $G \le GL(V)$ on $V$ and this induces a right action of $G$ on the dual of $V$ which can be extended to the symmetric algebra $F_q[V]$ by ring automorphisms. In this thesis we find the explicit generators and relations among these generators for the ring of invariants $F_q[V] G$. The main body of the research is in chapters 4, 5 and 6. In chapter 4, we consider three subgroups of the general linear group which preserve singular alternating, singular hermitian and singular quadratic forms respectively, and find rings of invariants for these groups. We then go on to consider, in chapter 5, a subgroup of the symplectic group. We take two special cases for this subgroup. In the first case we find the ring of invariants for this group. In the second case we progress to the ring of invariants for this group but the problem is still open. Finally, in chapter 6, we consider the orthogonal groups in even characteristic. We generalize some of the results of [24]. This generalization is important because it will help to calculate the rings of invariants of the orthogonal groups over any finite field of even characteristic.

This thesis is primarily concerned with saturated fusion systems over groups of shape q\(^r\) : q where q = p\(^n\) for some odd prime p and some natural number n. We shall present two results related to these fusion systems. Our first result is a complete classification of saturated fusion systems over a Sylow p-subgroup of SL\(_3\)(q) (which has shape q\(^3\) : q). This extends a result of Albert Ruiz and Antonio Viruel, who studied the case when q = p in [36]. As an immediate consequence of this result we shall have a complete classification of p-local finite groups over Sylow p-subgroups of SL\(_3\)(q). In the second half of this thesis we shall construct an infinite family of exotic fusion systems over some groups of shape p\(^r\) : p. This extends some work of Broto, Levi and Oliver, who studied the case when r = 3 in [12].