Abstract

McKay’s Monstrous E8 observation has provided further evidence, along with the evidence provided by the study of Monstrous Moonshine, that the Monster is intimately linked with a wide spectrum of other mathematical objects and, one might even say, with the natural organization of the universe. Although these links have been observed and facts about them proved, we have yet to understand exactly where and how they originate. We here review a set of conditions, due to Duncan, imposed on arithmetic subgroups of PSL2(R) that return McKay’s Monstrous E8 diagram. The purpose is to compare these with Conway, McKay and Sebbar’s (CMS) conditions that return the complete set of Monstrous Moonshine groups in order to gain some insight on their meaning. By way of doing this review of Duncan’s conditions, we will also review and elaborate on Conway’s method for understanding groups like Γ0(N).