Related papers: Power-associative, conjugacy closed loops

We show how to obtain all nonassociative Moufang loops of order less than 64 and 4262 nonassociative Moufang loops of order 64 in a unified way. We conjecture that there are no other nonassociative Moufang loops of order 64. The main idea…

$D$-loops are loops with the antiautomorphic inverse property. The class of such loops is larger than the class of IP-loops. The smallest $D$-loops which is not an IP-loop has six elements. We prove several basic properties of such loops…

The right conjugacy closed loops of order 2p, where p is an odd prime, are classified up to isomorphism.

Given a feature set for the shape of a closed loop, it is natural to ask which features in that set do not change when the starting point of the path is moved. For example, in two dimensions, the area enclosed by the path does not depend on…

Isotopes of C-loops with unique non-identity squares are shown to be both C-loops and A-loops. The relationship between C-loops and Steiner loops is further studied. Central loops with the weak and cross inverse properties are also…

It is well known that for a non pseudocompact space X, the family (X) of all intermediate subrings of C(X) which contain bounded real valued continuous functions contains at least 2c many distinct rings. We show that if in addition X is…

Let $L$ be a Moufang loop which is centrally nilpotent of class 2. We first show that the nuclearly-derived subloop (normal associator subloop) $L^*$ of $L$ has exponent dividing 6. It follows that $L_p$ (the subloop of $L$ of elements of…

We study ordered configuration spaces $C(n;p,q)$ of $n$ hard squares in a $p \times q$ rectangle, a generalization of the well-known "15 Puzzle". Our main interest is in the topology of these spaces. Our first result is to describe a…

The set of formal power series with coefficients in an associative but noncommutative algebra becomes a loop with the substitution product. We initiate the study of this loop by describing certain Lie and Sabinin algebras related to it.…

A loop is (right) automorphic if all its (right) inner mappings are automorphisms. Using the classification of primitive groups of small degrees, we show that there is no nonassociative simple commutative automorphic loop of order less than…

We say that a loop is unbreakable when it does not have nontrivial subloops. While the cyclic groups of prime order are the only unbreakable finite groups, we show that nonassociative unbreakable loops exist for every order n >= 5. We…

Using the standard representation of the Dirac equation we show that, up to signs, there exist only TWO SETS of consistent solutions for the matrices of charge conjugation (C), parity (P), and time reversal (T). In both cases, P^2=-1, and…

It is proved that the following conditions are equivalent for an infinite non-associative commutative Moufang loop $Q$: 1) $Q$ satisfies the minimum condition for subloops; 2) if the loop $Q$ contains a centrally solvable subloop of class…

Finite groups are of the greatest importance in science. Loops are a simple generalization of finite groups: they share all the group axioms except for the requirement that the binary operation be associative. The least loops that are not…

For most (and possibly all) non-associative finite simple Moufang loops, three generators of order 3 can be chosen so that each two of them generate a group isomorphic to $(3, 3 | 3, p)$. The subgroup structure of $(3, 3 | 3, p)$ depends on…

We establish a loop space decomposition for certain $CW$-complexes with a single top cell in the presence of a spherical pair, thereby generalizing several known decompositions of Poincar\'{e} duality complexes in which a loop of a product…

We classify Bol loops of order $27$, using a combination of theoretical results and computer search. There are $15$ Bol loops of order $27$, including five groups. New constructions for the ten nonassociative Bol loops of order $27$ are…

We define a variety of loops called semiautomorphic, inverse property loops that generalize Moufang and Steiner loops. We first show an equivalence between a previously studied variety of loops. Next we extend several known results for…

In the recently emerging field of nonabelian group-based cryptography, a prominently used one-way function is the Conjugacy Search Problem (CSP), and two important classes of platform groups are polycyclic and matrix groups. In this paper,…

A loop whose inner mappings are automorphisms is an \emph{automorphic loop} (or \emph{A-loop}). We characterize commutative (A-)loops with middle nucleus of index 2 and solve the isomorphism problem. Using this characterization and certain…