Searching for small simple automorphic loops

Johnson, Kenneth W.; Kinyon, Michael; Nagy, Gábor P.; Vojtěchovský, Petr

doi:10.1112/s1461157010000173

Cited by 11 publications

(13 citation statements)

References 30 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Since a loop Q is simple if and only if Mlt(Q) is a primitive permutation group on Q, we approach the problem from the direction of primitive groups. In [19] we proved computationally, using the library of primitive groups in GAP, that a finite simple automorphic loop of order less than 2500 is associative. Here we show that if Q is a finite simple nonassociative automorphic loop then the socle of Mlt(Q) is not regular, hence, by the O'Nan-Scott theorem, Mlt(Q) is of almost simple type, of diagonal type or of product type.…”

Section: Introductionmentioning

confidence: 99%

The structure of automorphic loops

Kinyon

Kunen

Phillips

et al. 2016

Trans. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

Abstract. Automorphic loops are loops in which all inner mappings are automorphisms. This variety of loops includes, for instance, groups and commutative Moufang loops.We study uniquely 2-divisible automorphic loops, particularly automorphic loops of odd order, from the point of view of the associated Bruck loops (motivated by Glauberman's work on uniquely 2-divisible Moufang loops) and the associated Lie rings (motivated by a construction of Wright). We prove that every automorphic loop Q of odd order is solvable, contains an element of order p for every prime p dividing |Q|, and |S| divides |Q| for every subloop S of Q.There are no finite simple nonassociative commutative automorphic loops, and there are no finite simple nonassociative automorphic loops of order less than 2500. We show that if Q is a finite simple nonassociative automorphic loop then the socle of the multiplication group of Q is not regular. The existence of a finite simple nonassociative automorphic loop remains open.Let p be an odd prime. Automorphic loops of order p or p 2 are groups, but there exist nonassociative automorphic loops of order p 3 , some with trivial nucleus (center) and of exponent p. We construct nonassociative "dihedral" automorphic loops of order 2n for every n > 2, and show that there are precisely p − 2 nonassociative automorphic loops of order 2p, all of them dihedral.

show abstract

Section: Introductionmentioning

confidence: 99%

The structure of automorphic loops

Kinyon

Kunen

Phillips

et al. 2016

Trans. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

show abstract

“…It is known that there are no simple nonassociative automorphic loops of order less than 2500 and no simple nonassociative commutative automorphic loops of order less than 2 12 [14]. The main result of this paper shows that in the commutative case, Problem 1 has a negative answer, and in fact, more than that.…”

Section: Introductionmentioning

confidence: 77%

Solvability of commutative automorphic loops

Grishkov

Kinyon

Nagy

2014

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…We defer the formal definition until Section 2, but note here that one defining axiom is commutativity. -loops include as special cases two classes of loops which have appeared in the literature: commutative Respects, Inverses, and Flexible (RIF) loops [15] and commutative automorphic loops [4,[10][11][12][13]. We will not discuss RIF loops any further in this paper, but we will review the notion of commutative automorphic loop in Section 2.…”

Section: Loops Categorically Isomorphic To Bruck Loops 3683mentioning

confidence: 98%

A Class of Loops Categorically Isomorphic to Bruck Loops of Odd Order

Greer

2014

Communications in Algebra

View full text Add to dashboard Cite

We define a new variety of loops, -loops. After showing -loops are power-associative, our main goal is showing a categorical isomorphism between Bruck loops of odd order and -loops of odd order. Once this has been established, we can use the well known structure of Bruck loops of odd order to derive the Odd Order, Lagrange and Cauchy Theorems for -loops of odd order, as well as the nontriviality of the center of finite -p-loops (p odd). Finally, we answer a question posed by Jedlička, Kinyon and Vojtěchovský about the existence of Hall -subloops and Sylow p-subloops in commutative automorphic loops.

show abstract

Searching for small simple automorphic loops

Cited by 11 publications

References 30 publications

The structure of automorphic loops

The structure of automorphic loops

Solvability of commutative automorphic loops

A Class of Loops Categorically Isomorphic to Bruck Loops of Odd Order

Contact Info

Product

Resources

About