Solvability of commutative automorphic loops

Grishkov, A.; Kinyon, Michael; Nagy, Gábor P.

doi:10.1090/s0002-9939-2014-12053-3

Cited by 6 publications

(8 citation statements)

References 21 publications

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“…The main open problem in the theory of automorphic loops is the existence or nonexistence of a nonassociative finite simple automorphic loop, cf., Problem 10.1. By Theorem 6.6 and by the main results of [14], such a loop would be noncommutative and of even order, though not a 2-loop.…”

Section: Finite Simple Automorphic Loopsmentioning

confidence: 88%

“…Although the classification of finite simple automorphic loops remains open, results from group theory about characteristic subgroups hold analogously for characteristic subloops of automorphic loops with essentially the same proofs (cf. the closing remarks of [14]). Part (ii) of the following result is [3, Thm.…”

Section: Finite Simple Automorphic Loopsmentioning

confidence: 94%

“…By [16, Thm. 6.2], Q has order a power of 2 and then by [14,Thm. 3], Q is solvable, a contradiction.…”

Section: Finite Simple Automorphic Loopsmentioning

confidence: 99%

“…By Corollary 4.8, Q has an element a of order 2. If every element of Q had order 2, then by [14,Thm. 8], Q itself would have order a power of 2, a contradiction.…”

Section: Dihedral Automorphic Loopsmentioning

confidence: 99%

“…In recent years, a detailed structure theory has emerged for commutative automorphic loops. For instance, the Odd Order, Lagrange and Cauchy Theorems hold for commutative automorphic loops, a finite commutative automorphic loop has order p k if and only if each element has order a power of p, and a finite commutative automorphic loop decomposes as a direct product of a loop of odd order and a loop of order a power of 2 [16]; there are no finite simple nonassociative commutative automorphic loops [14]; for an odd prime p, if Q is a finite commutative automorphic p-loop then Mlt(Q) is a p-group and Q is centrally nilpotent [5,18]; for an odd prime p, commutative automorphic loops of order p, p 2 , 2p, 2p 2 , 4p, 4p 2 are groups [17].…”

Section: Introductionmentioning

confidence: 99%

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The structure of automorphic loops

Kinyon

Kunen

Phillips

et al. 2016

Trans. Amer. Math. Soc.

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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.

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Section: Finite Simple Automorphic Loopsmentioning

confidence: 88%

Section: Finite Simple Automorphic Loopsmentioning

confidence: 94%

“…By [16, Thm. 6.2], Q has order a power of 2 and then by [14,Thm. 3], Q is solvable, a contradiction.…”

Section: Finite Simple Automorphic Loopsmentioning

confidence: 99%

“…By Corollary 4.8, Q has an element a of order 2. If every element of Q had order 2, then by [14,Thm. 8], Q itself would have order a power of 2, a contradiction.…”

Section: Dihedral Automorphic Loopsmentioning

confidence: 99%