Constructions of Commutative Automorphic Loops

Jedlička, Přemysl; Kinyon, Michael; Vojtěchovský, Petr

doi:10.1080/00927870903200877

Cited by 20 publications

(44 citation statements)

References 5 publications

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“…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

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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.

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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

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“…To get some insight into the problem, more constructions of commutative A-loops which are 2-loops are needed; see [11].…”

Section: Open Problemsmentioning

confidence: 99%

“…While every A-loop of prime order p is isomorphic to the cyclic group of order p, a class of nonassociative commutative A-loops of order pq (2 < p < q primes) was found by Drápal [7]. A survey of known constructions and the classification of commutative A-loops of small orders will appear in the planned sequel [11] to this paper. In [11], we will also give an example of a commutative A-loop of order 16 that is not centrally nilpotent.…”

Section: Introductionmentioning

confidence: 99%

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The structure of commutative automorphic loops

Jedlička

Kinyon

Vojtěchovský

2011

Trans. Amer. Math. Soc.

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Abstract. An automorphic loop (or A-loop) is a loop whose inner mappings are automorphisms. Every element of a commutative A-loop generates a group, and (xy) −1 = x −1 y −1 holds. Let Q be a finite commutative A-loop and p a prime. The loop Q has order a power of p if and only if every element of Q has order a power of p. The loop Q decomposes as a direct product of a loop of odd order and a loop of order a power of 2. If Q is of odd order, it is solvable. If A is a subloop of Q, then |A| divides |Q|. If p divides |Q|, then Q contains an element of order p. If there is a finite simple nonassociative commutative A-loop, it is of exponent 2.

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“…By [19,Proposition 6.1, Theorem 6.2], a finite simple commutative automorphic loop is either a cyclic group of prime order, or a loop of exponent two and order a power of two. It was shown in [18], by an exhaustive search with a finite model builder, that there are no simple non-associative commutative automorphic loops of order less than 32.…”

Section: Simple Automorphic Loopsmentioning

confidence: 99%

Searching for small simple automorphic loops

Johnson

Kinyon

Nagy

et al. 2011

LMS J. Comput. Math.

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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 non-associative simple commutative automorphic loop of order less than 2 12 , and no non-associative simple automorphic loop of order less than 2500. We obtain numerous examples of non-associative simple right automorphic loops. We also prove that every automorphic loop has the antiautomorphic inverse property, and that a right automorphic loop is automorphic if and only if its conjugations are automorphisms.

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Constructions of Commutative Automorphic Loops

Cited by 20 publications

References 5 publications

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

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

The structure of commutative automorphic loops

Searching for small simple automorphic loops

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