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We define the notion of adjustment for strict Lie 2-groups and provide the complete cocycle description for non-Abelian gerbes with connections whose structure 2-group is an adjusted 2-group. Most importantly, we depart from the common…

We study torsors over 2-groups and their morphisms. In particular, we study the first non-abelian cohomology group with values in a 2-group. Butterfly diagrams encode morphisms of 2-groups and we employ them to examine the functorial…

We show that the endomorphisms of a compact connected group that extend to endomorphisms of every compact overgroup are precisely the trivial one and the inner automorphisms; this is an analogue, for compact connected groups, of results due…

We construct an extension of the Poincare group which involves a mixture of internal and space-time supersymmetries. The resulting group is an extension of the superPoincare group with infinitely many generators which carry internal and…

For a CSA group $G$ and a wide class of abelian groups $A$ we give an explicit construction for the tensor $A$-completion of $G$ using free products with amalgamations. We apply the obtained results to the study of basic properties of…

The paper investigates two invariants for totally disconnected locally compact groups: the number of ends and the rational discrete cohomological dimension. For such a compactly generated group $G$ it is shown that its number of ends can be…

A difference matrix over a group is a discrete structure that is intimately related to many other combinatorial designs, including mutually orthogonal Latin squares, orthogonal arrays, and transversal designs. Interest in constructing…

In this paper we construct abelian extensions of the group of diffeomorphisms of a torus. We consider the jacobian map, which is a crossed homomorphism from the group of diffeomorphisms into a toroidal gauge group. A pull-back under this…

The notion of essential submodules and essential extensions of modules are extended to groups (typically nonabelian), and several necessary and sufficient conditions for a group to possess a proper essential subgroup are investigated.…

Suppose V is a finite dimensional, complex vector space, A is a finite set of codimension one subspaces of V, and G is a finite subgroup of the general linear group GL(V) that permutes the hyperplanes in A. In this paper we study invariants…

For a unital ring $S$, an $S$-linear quasigroup is a unital $S$-module, with automorphisms $\rho$ and $\lambda$ giving a (nonassociative) multiplication $x\cdot y=x^\rho+y^\lambda$. If $S$ is the field of complex numbers, then ordinary…

We use the theory of canonical models of Shimura varieties to describe the projective limit of the curves Y(N), all N, and its automorphism group. In particular we prove that the Galois group of Q(CM) over Q is an extension of a certain…

We prove the "divisible case" of the Milnor-Bloch-Kato conjecture (which is the first step of Voevodsky's proof of this conjecture for arbitrary prime l) in a rather clear and elementary way. Assuming this conjecture, we construct a 6-term…

We study the closed group of homeomorphisms of the boundary of real hyperbolic space generated by a cocompact Kleinian group $G_1$ and a quasiconformal conjugate $h^{-1}G_2 h$ of a cocompact group $G_2$. We show that if the conjugacy $h$ is…

Quandle colorings and cocycle invariants are studied for composite knots, and applied to chirality and abelian extensions. The square and granny knots, for example, can be distinguished by quandle colorings, so that a trefoil and its mirror…

This is the second installment of an exposition of an ACL2 formalization of finite group theory. The first, which was presented at the 2022 ACL2 workshop, covered groups and subgroups, cosets, normal subgroups, and quotient groups,…

Let $p$ be an odd prime. Denote a Sylow $p$-subgroup of $GL_2(\mathbb{Z}/p^n)$ and $SL_2(\mathbb{Z}/p^n)$ by $S_p(n,GL)$ and $S_p(n,SL)$ respectively. The theory of stable elements tells us that the mod-$p$ cohomology of a finite group is…

For line arrangements in P^2 with nice combinatorics (in particular, for those which are nodal away the line at infinity), we prove that the combinatorics contains the same information as the fundamental group together with the meridianal…

The idea that the cohomology of finite groups might be fruitfully approached via the cohomology of ambient semisimple algebraic groups was first shown to be viable in the papers [CPS75] and [CPSvdK77]. The second paper introduced, through a…

We construct an extension $E(A,G)$ of a given group $G$ by infinite non-Archimedean words over an discretely ordered abelian group like $Z^n$. This yields an effective and uniform method to study various groups that "behave like $G$". We…