Every diassociative A-loop is Moufang

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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“…5]. The general case was finally settled in [20]: Every diassociative automorphic loop is a Moufang loop.…”

Section: Introductionmentioning

confidence: 99%

The structure of automorphic loops

Kinyon

Kunen

Phillips

et al. 2016

Trans. Amer. Math. Soc.

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“…The following statements show an essential difference between A-loops and Moufang loops. Proposition 1.5 [12]. For an A-loop L, the following are equivalent:…”

Section: Preliminariesmentioning

confidence: 99%

On identities in centrally nilpotent Moufang loops and centrally nilpotent A-loops

Sandu

2014

Preprint

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“…To see what happens, let us look at a concrete example, which is essentially the main theorem of [4]. Here is what the assumptions look like in PROVER9 syntax.…”

Section: The First Proof An Atp Finds Is Usually Very Complex and Very Unstable!mentioning

confidence: 99%