What is the monster?

We explain a conjecture relating the monster simple group to an algebraic variety that was discovered in a non-monstrous context.

This article is a short and elementary introduction to the monstrous moonshine aiming to be as accessible as possible. I first review the classification of finite simple groups out of which the monster naturally arises, and features of the latter that are needed in order to state the moonshine conjecture of Conway and Norton. Then I motivate modular functions and modular forms from the classification of complex tori, with the definitions of the J-invariant and its q-expansion...

We discuss ways in which the black-box model for computation is or is not applicable to the Monster sporadic simple group. Conversely, we consider whether methods of computation in the Monster can be generalised to other situations, for example to groups of `cross-characteristic' type.

This is an introduction to finite simple groups, in particular sporadic groups, intended for physicists. After a short review of group theory, we enumerate the $1+1+16=18$ families of finite simple groups, as an introduction to the sporadic groups. These are described next, in three levels of increasing complexity, plus the six isolated "pariah" groups. The (old) five Mathieu groups make up the first, smallest order level. The seven groups related to the Leech lattice, includ...

This is a book on Group.

We describe various approaches to constructing groups which may serve as Lie group analogs for the monster Lie algebra of Borcherds.

We use uniqueness of a VOA (vertex operator algebra) extension of $(V_{EE_8}^+)^3$ to a Moonshine type VOA to give a new existence proof of a finite simple group of Monster type. The proof is relatively direct. Our methods depend on VOA representation theory and are free of many special calculations which traditionally occur in theory of the Monster.

We describe the collection of finite simple groups, with a view on physical applications. We recall first the prime cyclic groups $Z_p$, and the alternating groups $Alt_{n>4}$. After a quick revision of finite fields $\mathbb{F}_q$, $q = p^f$, with $p$ prime, we consider the 16 families of finite simple groups of Lie type. There are also 26 \emph{extra} "sporadic" groups, which gather in three interconnected "generations" (with 5+7+8 groups) plus the Pariah groups (6). We poi...

The commencement of monstrous moonshine is a connection between the largest sporadic simple group---the monster---and complex elliptic curves. Here we explain how a closer look at this connection leads, via the Thompson group, to recently observed relationships between the non-monstrous sporadic simple group of O'Nan and certain families of elliptic curves defined over the rationals. We also describe umbral moonshine from this perspective.

We describe computer calculations that were used in 2016 to classify subgroups of the Monster isomorphic to $PSL_2(8)$, containing $7B$-elements. It turns out that there is no such $PSL_2(8)$ in the Monster. These calculations confirm earlier unpublished calculations by P. E. Holmes that obtained the same result. The result has also been confirmed in independent calculations by H. Dietrich, M. Lee and T. Popiel, using different software by M. Seysen. Thus this experimental re...