Related papers: The structure of extra loops

A finite group G is exceptional if it has a quotient Q whose minimal faithful permutation degree is greater than that of G. We say that Q is a distinguished quotient. The smallest examples of exceptional p-groups have order p^5. For an odd…

A practical implementation of the non-Abelian Stokes theorem for topologically nontrivial loops (knots) with possible intersections is proposed.

Let $q$ be an odd prime power and suppose that $a,b\in\mathbb{F}_q$ are such that $ab$ and $(1{-}a)(1{-}b)$ are nonzero squares. Let $Q_{a,b} = (\mathbb{F}_q,*)$ be the quasigroup in which the operation is defined by $u*v=u+a(v{-}u)$ if…

We compute equations for the families of elliptic curves 9-congruent to a given elliptic curve. We use these to find infinitely many non-trivial pairs of 9-congruent elliptic curves over Q, i.e. pairs of non-isogenous elliptic curves over Q…

The Cayley-Dickson algebras R (real numbers), C (complex numbers), H (quaternions), O (octonions), S (sedenions), and T (trigintaduonions) have attracted the attention of several mathematicians and physicists because of their important…

Non-local twist operators are introduced for the O(n) and Q-state Potts models in two dimensions which, in the limits n -> 0 (resp. Q -> 1) count the numbers of self-avoiding loops (resp. percolation clusters) surrounding a given point.…

Automorphic loops are loops in which all inner mappings are automorphisms. This variety of loops includes groups and commutative Moufang loops. A half-isomorphism $f : G \longrightarrow K$ between multiplicative systems $G$ and $K$ is a…

The irreducible tensor bases of exceptional Lie algebras G2, F4 and E6 are built by grouping their Cartan-Weyl bases according to the respective chains G2> SO(3) * SO(3), F4 > SO(3)*SO(3)*SO(3)*SO(3) and E6> SO(3)*SO(3)*SO(3)*SO(3). The…

We give an explicit description of the category of central extensions of a group scheme by a sheaf of Abelian groups. Based on this, we describe a framework for computing with central extensions of finite commutative group schemes, torsors…

For every prime $p$, we construct an infinite countable group that contains precisely $p-1$ elements which are not $p$th powers.

We show that requiring sixteen supersymmetries in quantum mechanical gauge theory implies the existence of a web of constrained interactions. Contrary to conventional wisdom, these constraints extend to arbitrary orders in the momentum…

It is proved that the maximum condition for subloops in a commutative Moufang loop $Q$ is equivalent with the conditions of finite generating of different subloops of the loop $Q$ and different subgroups of the multiplication group of the…

We introduce a class of non-Moufang loops satisfying the Moufang's theorem.

Finite nonassociative division algebras (i.e., finite semifields) with 243 elements are completely classified.

For a prime p, we study the Galois groups of maximal pro-$p$ extensions of imaginary quadratic fields unramified outside a finite set $S$, where $S$ consists of one or two finite places not lying above $p$. When $p$ is odd, we give explicit…

A long-standing conjecture asserts that every finite nonabelian $p$-group has a non-inner automorphism of order $p$. In this paper we prove the conjecture for finite $p$-groups of coclass $4$ and coclass $5$ ($p\ge 5$). We also prove the…

We describe a method for counting the number of extensions of $\mathbb{Q}_p$ with a given Galois group $G$, founded upon the description of the absolute Galois group of $\mathbb{Q}_p$ due to Jannsen and Wingberg. Because this description is…

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 identify a structural pattern in the construction of known infinite families of trees whose independence polynomials are not log-concave. Using this pattern and properties of polynomial ring ideals, we derive linear recurrences for these…

This book introduces several new classes of groupoid, like polynomial groupoids, matrix groupoids, interval groupoids,polynomial interval groupoids, matrix interval groupoids and their neutrosophic analogues. Interval groupoid happens to be…