A Class of Simple Moufang Loops

Paige, Lowell J.

doi:10.2307/2032757

Cited by 17 publications

(23 citation statements)

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“…L. Paige discovered an infinite family of nonassociative finite simple Moufang loops [22], and M. Liebeck proved that no other nonassociative finite simple Moufang loops exist [14]. Although examples of finite simple non-Moufang Bol loops were not easy to find, they abound [18] even in the more restrictive case of Bruck loops [2,17].…”

Section: Introductionmentioning

confidence: 99%

Bol loops and Bruck loops of order pq

2017

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Right Bol loops are loops satisfying the identity ((zx)y)x = z((xy)x), and right Bruck loops are right Bol loops satisfying the identity (xy) −1 = x −1 y −1 . Let p and q be odd primes such that p > q. Advancing the research program of Niederreiter and Robinson from 1981, we classify right Bol loops of order pq. When q does not divide p 2 − 1, the only right Bol loop of order pq is the cyclic group of order pq. When q divides p 2 − 1, there are precisely (p − q + 4)/2 right Bol loops of order pq up to isomorphism, including a unique nonassociative right Bruck loop Bp,q of order pq.Let Q be a nonassociative right Bol loop of order pq. We prove that the right nucleus of Q is trivial, the left nucleus of Q is normal and is equal to the unique subloop of order p in Q, and the right multiplication group of Q has order p 2 q or p 3 q. When Q = Bp,q, the right multiplication group of Q is isomorphic to the semidirect product of Zp × Zp with Zq. Finally, we offer computational results as to the number of right Bol loops of order pq up to isotopy.2010 Mathematics Subject Classification. Primary: 20N05; Secondary: 12F05, 15B05, 15B33, 20D20. Key words and phrases. Bol loop, Bruck loop, K-loop, Bol loop of order pq, Bruck loop of order pq, multiplication group, nucleus, dihedral group, twisted subgroup, eigenvalue of a circulant matrix, quadratic field extension.Michael K.

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Section: Introductionmentioning

confidence: 99%

Bol loops and Bruck loops of order pq

2017

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“…See Doro [7] for recent results on simple Moufang loops. Recently, Liebeck [12] has shown, by using the classification of finite simple groups, that there is no finite simple non-associative (non-group) Moufang loop other than the following loops M*(q), defined by Paige [13] for any prime power q. Define = 1) a ,beGF(q), a, Pe (GFfe)) 3 where ° and x denote the scalar and vector products in (GF(q)) 3 In § 2, we will determine the character tables of M(q) and M*(q).…”

Section: Introductionmentioning

confidence: 99%

The Character Tables of Paige's Sample Moufang Loops and Their Relationship to the Character Tables of Psl(2, q )

Bannaĭ

Song

1989

Proceedings of the London Mathematical Society

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“…[10], [50,Ch.2], [25]). Over the field R there are two nonisomorphic octonion algebras: the classical division Cayley octonions Ø and the split matrix Cayley-Dickson algebra ZornpRq [10], also known as the Zorn vector-matrix algebra [38]. The last one may be defined over an arbitrary commutative ring.…”

Section: Loop Of Unitary Elementsmentioning

confidence: 99%

Loop of formal diffeomorphisms and Faà di Bruno coloop bialgebra

Frabetti

Shestakov

2019

Advances in Mathematics

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We consider a generalization of (pro)algebraic loops defined on general categories of algebras and the dual notion of a coloop bialgebra suitable to represent them as functors. We prove that the natural loop of formal diffeomorphisms with associative coefficients is proalgebraic, and we give the closed formulas of the codivisions on its coloop bialgebra. This result provides a generalization of the Lagrange inversion formula to series with non-commutative coefficients, and a loop-theoretic explanation to the existence of the non-commutative Faà di Bruno Hopf algebra. MSC: 20N05, 14L17, 18D35, 16T30Inv qnd Diff and which turn out to be extremely efficient in handling the combinatorial content of renormalization procedures [6,7,8,47,43].The toy model φ 3 theory used by Connes-Kreimer is a scalar field theory and leads to the commutative algebra A " C of amplitudes. However, interesting physical situations involve noncommutative algebras. In fact, Feynman amplitudes are complex numbers for single scalar fields, the coupling constants and the renormalization factors, but they are 4ˆ4 complex matrices for the fermionic or bosonic fields, and may be represented by higher order matrices for theories involving several interacting fields. In this case, forcing the final counterterms to be scalar, as imposed by the fact that the renormalization factors act on the (scalar) Lagrangian, prevents us from describing the renormalization in a functorial way, as shown by the results in [48], where the Hopf algebra does not represent a functorial group on A " M 4 pCq. In order to preserve this functoriality, there is a need to understand Dyson's formulas for sets of series InvpAq and DiffpAq also when A is not a commutative algebra. This is the motivation for the present work.

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A Class of Simple Moufang Loops

Cited by 17 publications

References 2 publications

Bol loops and Bruck loops of order pq

Bol loops and Bruck loops of order pq

The Character Tables of Paige's Sample Moufang Loops and Their Relationship to the Character Tables of Psl(2, q )

Loop of formal diffeomorphisms and Faà di Bruno coloop bialgebra

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