Related papers: Moonshine for All Finite Groups

Let $G$ be a finite group. If $M_n < M_{n-1} < \ldots < M_1 < M_{0}=G $ where $M_i$ is a maximal subgroup of $M_{i-1}$ for all $i=1, \ldots ,n$, then $M_n $ ($n > 0$) is an \emph{$n$-maximal subgroup} of $G$. A subgroup $M$ of $G$ is called…

We describe a family of indefinite theta functions of signature $(1,1)$ that can be expressed in terms of trace functions of vertex algebras built from cones in lattices. The family of indefinite theta functions considered has interesting…

We describe a procedure for constructing a generalized Thompson group out of a family of groups that is equipped with what we call a cloning system. The previously known Thompson groups F, V, Vbr and Fbr arise from this procedure using,…

Let $G$ be a finite group and let $p$ be a prime. In this paper, we study the structure of finite groups with a large number of $p$-regular conjugacy classes or, equivalently, a large number of irreducible $p$-modular representations. We…

We show interesting relations between extremal partition functions of a family of conformal field theories and dimensions of the irreducible representations of the Fischer-Griess Monster sporadic group. We argue that these relations can be…

Let G be a finite subgroup of GL_n(C). G-constellations are a scheme-theoretic generalization of orbits of G in C^n. We study flat families of G-constellations parametrised by an arbitrary resolution of the quotient space C^n/G. We develop…

A remarkable result of Thompson states that a finite group is soluble if and only if its two-generated subgroups are soluble. This result has been generalized in numerous ways, and it is in the core of a wide area of research in the theory…

We prove that for any prime $\ell$, any finite group has as many irreducible complex characters of degree prime to $\ell$ as the normalizers of its Sylow $\ell$-subgroups. This equality was conjectured by John McKay. The conjecture was…

This paper continues the study of generalized amalgamation properties. Part of the paper provides a finer analysis of the groupoids that arise from failure of 3-uniqueness in a stable theory. We show that such groupoids must be abelian and…

Let R be a ring and G a group. An R-module A is said to be artinian-by-(finite rank) if TorR(A) is artinian and A/TorR(A) has finite R-rank. The authors study ZG-modules A such that A/CA(H) is artinian-by-(finite rank) (as a Z-module) for…

We construct two families of refinements of the (projectivized) support variety of a finite dimensional module $M$ for a finite group scheme $G$. For an arbitrary finite group scheme, we associate a family of {\it non maximal rank…

Let R be any ring (with 1), \Gamma a group and R\Gamma the corresponding group ring. Let H be a subgroup of \Gamma of finite index. Let M be an R\Gamma -module, whose restriction to RH is projective. Moore's conjecture: Assume for every…

Over any partially ordered abelian group whose positive cone is closed in an appropriate sense and has finitely many faces, modules that satisfy a weak finiteness condition admit finite primary decompositions. This conclusion rests on the…

We show that the modular group has an infinite family of finite index subgroups, each of which has the same trace set as the modular group itself. Various congruence subgroups of the modular group, and the Bianchi groups, are also shown to…

Let $G$ be an abelian group of order $n$ and let $R$ be a commutative ring which admits a homomorphism ${\Bbb Z}[\zeta_{n}]\ra R$, where $\zeta_{n}$ is a (complex) primitive $n$-th root of unity. Given a finite $R[G\e]$-module $M$, we…

We establish several new bounds for the number of conjugacy classes of a finite group, all of which involve the maximal number c of conjugacy classes of a normal subgroup fixed by some element of a suitable subset of the group. To apply…

Consider the Mackey functor assigning to each finite group G the Green ring of finitely generated kG-modules, where k is a field of characteristic p>0. Thevenaz foresaw in 1988 that the class of primordial groups for this functor is the…

The D1-D5-KK-p system naturally provides an infinite dimensional module graded by the dyonic charges whose dimensions are counted by the Igusa cusp form, Phi_{10}(Z)$. We show that the Mathieu group, M_{24}, acts on this module by…

In Monstrous moonshine, genus 0 property and the notion of replicability are strongly connected. With regards to recent developments of moonshine, we investigate a higher genus generalization of replicability for a general automorphic form.…

In this paper we address the following two closely related questions. First, we complete the classification of finite symmetry groups of type IIA string theory on $K3\times \mathbb R^6$, where Niemeier lattices play an important role. This…