The structure of commutative automorphic loops

Jedlička, Přemysl; Kinyon, Michael; Vojtěchovský, Petr

doi:10.1090/s0002-9947-2010-05088-3

Cited by 21 publications

(49 citation statements)

References 15 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: 99%

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: 99%

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

Greer

2014

Communications in Algebra

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“…The foundations of the theory of commutative automorphic loops were laid by Jedlička et al [19], with such structural results such as the Cauchy theorem, Lagrange theorem, odd order theorem, etc. A paper analogous to [19], but without the assumption of commutativity, is in preparation [21]. By [19,Theorems 5.1,5.3,7.1], every finite commutative automorphic loop is a direct product of a solvable loop of odd order and a loop of order a power of two.…”

Section: Simple Automorphic Loopsmentioning

confidence: 99%

“…A paper analogous to [19], but without the assumption of commutativity, is in preparation [21]. By [19,Theorems 5.1,5.3,7.1], every finite commutative automorphic loop is a direct product of a solvable loop of odd order and a loop of order a power of two. 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.…”

Section: Simple Automorphic Loopsmentioning

confidence: 99%

“…By [19,Theorems 5.1,5.3,7.1], every finite commutative automorphic loop is a direct product of a solvable loop of odd order and a loop of order a power of two. 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%

“…[19], Problem 1.3 has a negative answer if and only if every finite commutative automorphic loop is solvable. Problem 1.4.…”

Section: Open Problemsmentioning

confidence: 99%

See 2 more Smart Citations

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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Incidence properties of cosets in loops

Kinyon¹,

Pula²,

Vojtěchovský³

2012

J of Combinatorial Designs

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Abstract:We study incidence properties among cosets of infinite loops, with emphasis on well-structured varieties such as antiautomorphic loops and Bol loops. While cosets in groups are either disjoint or identical, we find that the incidence structure in general loops can be much richer. Every symmetric design, for example, can be realized as a canonical collection of cosets of a infinite loop. We show that in the variety of antiautomorphic loops the poset formed by set inclusion among intersections of left cosets is isomorphic to that formed by right cosets. We present an algorithm that, given a infinite Bol loop S, can in some cases determine whether |S| divides | Q| for all infinite Bol loops Q with S ≤ Q, and even whether there is a selection of left cosets of S that partitions Q. This method results in a positive confirmation of Lagrange's Theorem for Bol loops for a few new cases of subloops. Finally, we show that in a left automorphic Moufang loop Q (in particular, in a commutative Moufang loop Q), two left cosets of S ≤ Q are either disjoint or they intersect in a set whose cardinality equals that of some subloop of S. q

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The structure of commutative automorphic loops

Cited by 21 publications

References 15 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

Searching for small simple automorphic loops

Incidence properties of cosets in loops

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