Odd Characterizations of Almost Simple Groups (PhD Thesis 2009)

In this PhD thesis we discuss methods of recognizing finite groups by the structure of normalizers of certain 3-subgroups. We explain a method for characterizing groups using character theoretic and block theoretic methods and we use these methods to characterize $\text{alt}(9)$. Furthermore, we describe a particular hypothesis related to the 3-local structure of finite groups of local characteristic 3 and characterize two almost simple proper extensions of ${\mathrm{\Omega }}_8^{+}(2)$ as examples of the local approach to group recognition. We also give a 3-local characterization of the sporadic simple group $\text{HN}$ using local methods whilst applying a character theoretic result.