Research
I'm mainly interested in the subgroup structure and representation theory of finite groups. Much of my work thus far has been on either generation problems or cohomology in finite simple groups.
Generation in finite simple groups
It is well known that a finite simple group may be generated by just two elements, and in fact it is very easy to do so: given a finite simple group \(G\), let \(P(G)\) denote the probability that two elements chosen uniformly at random from G generate the group. Then \(P(G) \to 1\) as \(\abs{G} \to \infty\), and in general \(P(G) \geq \frac{53}{90}\). It makes sense, then, to go further. In early work during my PhD, I investigated the generating graph of \(\PSL_2(q)\): a graph which encodes the generation structure of the group by taking as vertices the non-identity elements and connecting \(g\), \(h \in G\) with an edge whenever \(G = \gen{g, h}\).
In more recent work with Michael Giudici, we are currently investigating which finite simple groups act 4-arc-transitively on cubic (3-regular) graphs. This is, actually, a question about generation of finite simple groups since any group which acts 4-arc-transitively on some cubic graph can be generated by subgroups isomorphic to \(S_4\) and either \(D_{16}\) or \(SD_{16}\) which intersect in a \(D_8 \in \Syl_2 (S_4)\). We are currently working on showing that all sufficiently large linear groups in characteristic at least 5 have this property.
Group cohomology
Group cohomology is intricately linked with the extension theory of finite groups. The first group cohomology \(\H^1(G,V)\) classifies conjugacy classes of complements to \(V\) in \(V \rtimes G\) and the second cohomology group \(\H^2(G,V)\) classifies group extensions of \(G\) by \(V\). In the 80s, Guralnick conjectured that for a finite simple group \(G\) we would have that \(\dim \H^1(G,V) \leq c\) for some absolute constant \(c\), and in fact he conjectured that \(c = 2\) would suffice. Work of Lübeck has shown that this is unlikely to be true, yet no family has yet been shown to contradict the conjecture.
I spent my PhD investigating the cohomology of \(\PSL_2(q)\) in cross characteristic (\(r \not \mid q\)) and since then have determined \(\dim \H^n(G,V)\) for all irreducible representations \(V\) of \(\PSL_2(q)\) in cross characteristic, and similarly for other rank 1 groups of Lie type with cyclic Sylow \(r\)-subgroups. In future, I would like to obtain a full answer for this question for at least the rank 1 groups of Lie type, though \(\PSU_3(q)\) is likely to prove problematic when \(r \mid q+1\) as the representation type in this case is in fact wild.
Products of conjugacy classes in finite groups
Work in progress with Chris Parker.