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Loops and semidirect products
, 2000
"... A left loop (B, ·) is a set B together with a binary operation · such that (i) for each a ∈ B, the left translation mapping La: B → B defined by La(x) = a · x is a bijection, and (ii) there exists a twosided identity 1 ∈ B satisfying 1 · x = x · 1 = x for every x ∈ B. A right loop is similarly def ..."
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Cited by 7 (4 self)
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A left loop (B, ·) is a set B together with a binary operation · such that (i) for each a ∈ B, the left translation mapping La: B → B defined by La(x) = a · x is a bijection, and (ii) there exists a twosided identity 1 ∈ B satisfying 1 · x = x · 1 = x for every x ∈ B. A right loop is similarly defined, and a loop is both a right loop
Involutory decomposition of groups into twisted subgroups and subgroups
 J. Group Theory
, 2000
"... Gyrogroups are generalized groups modelled on the Einstein groupoid of all relativistically admissible velocities with their Einstein’s velocity addition as a binary operation. Einstein’s gyrogroup fails to form a group since it is nonassociative. The breakdown of associativity in the Einstein addit ..."
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Cited by 7 (2 self)
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Gyrogroups are generalized groups modelled on the Einstein groupoid of all relativistically admissible velocities with their Einstein’s velocity addition as a binary operation. Einstein’s gyrogroup fails to form a group since it is nonassociative. The breakdown of associativity in the Einstein addition does not result in loss of mathematical regularity owing to the presence of the relativistic effect known as the Thomas precession which, by abstraction, becomes an automorphism called the Thomas gyration. The Thomas gyration turns out to be the missing link that gives rise to analogies shared by gyrogroups and groups. In particular, it gives rise to the gyroassociative and the gyrocommuttive laws that Einstein’s addition possesses, in full analogy with the associative and the commutative laws that vector addition possesses in a vector space. The existence of striking analogies shared by gyrogroups
The Structure of Automorphism Groups of Cayley Graphs and Maps.
 J. Algebraic Combin
, 1998
"... The automorphism groups Aut(C(G; X)) and Aut(CM(G;X; p)) of a Cayley graph C(G;X) and a Cayley map CM(G;X; p) both contain an isomorphic copy of the underlying group G acting via left translations. In our paper, we show that both automorphism groups are rotary extensions ([6]) of the group G by the ..."
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Cited by 1 (0 self)
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The automorphism groups Aut(C(G; X)) and Aut(CM(G;X; p)) of a Cayley graph C(G;X) and a Cayley map CM(G;X; p) both contain an isomorphic copy of the underlying group G acting via left translations. In our paper, we show that both automorphism groups are rotary extensions ([6]) of the group G by the stabilizer subgroup of the vertex 1 G . We use this description to derive necessary and sufficient conditions to be satisfied by a finite group in order to be the (full) automorphism group of a Cayley graph or map and classify all the finite groups that can be represented as the (full) automorphism group of some Cayley graph or map. 1 Introduction and preliminaries The only graphs considered in this paper are finite Cayley graphs \Gamma = C(G; X) which are finite simple graphs defined for any finite group G and a set of generators X ae G with the property 1 G 62 X and x \Gamma1 2 X for each x 2 X. The set V (\Gamma) of vertices of the Cayley graph \Gamma = C(G; X) is the set of elements...
(1.1)
, 2000
"... Abstract. On the unit sphere S in a real Hilbert space H, we derive a binary operation ⊙ such that (S, ⊙) is a powerassociative Kikkawa left loop with twosided identity e0, i.e., it has the left inverse, automorphic inverse, and Al properties. The operation ⊙ is compatible with the symmetric space ..."
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Abstract. On the unit sphere S in a real Hilbert space H, we derive a binary operation ⊙ such that (S, ⊙) is a powerassociative Kikkawa left loop with twosided identity e0, i.e., it has the left inverse, automorphic inverse, and Al properties. The operation ⊙ is compatible with the symmetric space structure of S. (S, ⊙) is not a loop, and the right translations which fail to be injective are easily characterized. (S, ⊙) satisfies the left power alternative and left Bol identities “almost everywhere ” but not everywhere. Left translations are everywhere analytic; right translations are analytic except at −e0 where they have a nonremovable discontinuity. The orthogonal group O(H) is a semidirect product of (S, ⊙) with its automorphism group. The left loop structure of (S, ⊙) gives some insight into spherical geometry.
(1.1)
, 1999
"... Abstract. On the unit sphere S in a real Hilbert space H, we derive a binary operation ⊙ such that (S, ⊙) is a powerassociative Kikkawa left loop with twosided identity e0, i.e., it has the left inverse, automorphic inverse, and Al properties. The operation ⊙ is compatible with the symmetric space ..."
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Abstract. On the unit sphere S in a real Hilbert space H, we derive a binary operation ⊙ such that (S, ⊙) is a powerassociative Kikkawa left loop with twosided identity e0, i.e., it has the left inverse, automorphic inverse, and Al properties. The operation ⊙ is compatible with the symmetric space structure of S. (S, ⊙) is not a loop, and the right translations which fail to be injective are easily characterized. (S, ⊙) satisfies the left power alternative and left Bol identities “almost everywhere” but not everywhere. Left translations are everywhere analytic; right translations are analytic except at −e0 where they have an essential discontinuity. The orthogonal group O(H) is a semidirect product of (S, ⊙) with its automorphism group. The left loop structure of (S, ⊙) gives some insight into spherical geometry.
unknown title
, 2000
"... Nonassociative geometry and discrete structure of spacetime Abstract: A new mathematical theory, nonassociative geometry, providing a unified algebraic description of continuous and discrete spacetime, is introduced. ..."
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Nonassociative geometry and discrete structure of spacetime Abstract: A new mathematical theory, nonassociative geometry, providing a unified algebraic description of continuous and discrete spacetime, is introduced.