G-Loops and Permutation Groups

Kunen, Kenneth

doi:10.1006/jabr.1999.8006

Cited by 14 publications

(11 citation statements)

References 13 publications

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Order By: Relevance

“…As shown in [17], if q is prime, then every G-loop of prime order is a cyclic group; a similar result for order 3p, where p > 3 is prime, was established in [10]. On the other hand, non-group G-loops are known to exist for all even orders larger than 5 and all orders divisible by p 2 for some p > 2 [6].…”

Section: A Connection Between the Isotopically Transitive Latin Squarmentioning

confidence: 85%

Constructions of transitive latin hypercubes

Krotov

Potapov

2016

European Journal of Combinatorics

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A function f : {0, ..., q − 1} n → {0, ..., q − 1} invertible in each argument is called a latin hypercube. A collection (π 0 , π 1 , ..., π n ) of permutations of {0, ..., q − 1} is called an autotopism of a latin hypercube f if π 0 f (x 1 , ..., x n ) = f (π 1 x 1 , ..., π n x n ) for all x 1 , ..., x n . We call a latin hypercube isotopically transitive (topolinear) if its group of autotopisms acts transitively (regularly) on all q n collections of argument values. We prove that the number of nonequivalent topolinear latin hypercubes growths exponentially with respect to √ n if q is even and exponentially with respect to n 2 if q is divisible by a square. We show a connection of the class of isotopically transitive latin squares with the class of G-loops, known in noncommutative algebra, and establish the existence of a topolinear latin square that is not a group isotope. We characterize the class of isotopically transitive latin hypercubes of orders q = 4 and q = 5.

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Section: A Connection Between the Isotopically Transitive Latin Squarmentioning

confidence: 85%

Constructions of transitive latin hypercubes

Krotov

Potapov

2016

European Journal of Combinatorics

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“…This was applied in the case of loop theory in [7,8,9,10,11], and in the present paper, where much of the argument is cast in the spirit of Bruck and Paige [2], emphasizing grouptheoretic properties of the R(x) and L(x), rather than equations in the loop product and inverse. For example, in Corollary 4, the statement C(x, z) 3 = I conveys more information to most human readers than does the equivalent equation, z −1 (z((z −1 (z((z −1 (z(yx)x −1 ))x)x −1 ))x)x −1 ) = y, which might (in its ascii form) be a typical line of OTTER output.…”

Section: Computer-aided Proofsmentioning

confidence: 99%

Every diassociative A-loop is Moufang

Kinyon

Kunen

Phillips

2001

Proc. Amer. Math. Soc.

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“…for all x, y P Q, i.e., pθ, R c θ, R c θq is an autotopy a quasigroup pQ, ¨q. The element c is called a companion of θ [11,9,13].…”

Section: Introductionmentioning

confidence: 99%

Some properties of Neumann quasigroups

Didurik¹,

Shcherbacov²

2018

Preprint

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Any Neumann quasigroup pQ, ¨q (quasigroup with Neumann identity x ¨pyz ÿxq " z is called Neumann quasigroup) can be presented in the form x ¨y " x ´y, where pQ, `q is an abelian group. Automorphism group of Neumann quasigroup coincides with the group AutpQ, `q. Any Schweizer quasigroup (quasigroup with Schweizer identity xy ¨xz " zy is called Schweizer quasigroup) is a Neumann quasigroup and vice versa. Any Neumann quasigroup is a GA-quasigroup.

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G-Loops and Permutation Groups

Cited by 14 publications

References 13 publications

Constructions of transitive latin hypercubes

Constructions of transitive latin hypercubes

Every diassociative A-loop is Moufang

Some properties of Neumann quasigroups

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