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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 ..."
Abstract

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
Structural Interactions Of Conjugacy Closed Loops
"... We study conjugacy closed loops by means of their multi plication groups. Let Q be a conjugacy closed loop, N its nucleus, A the associator subloop, and L and R the left and right multiplication group, respectively. Put M = {a ∈ Q; La ∈ R}. We prove that the cosets of A agree with orbits of [L, R], ..."
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Cited by 6 (2 self)
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We study conjugacy closed loops by means of their multi plication groups. Let Q be a conjugacy closed loop, N its nucleus, A the associator subloop, and L and R the left and right multiplication group, respectively. Put M = {a ∈ Q; La ∈ R}. We prove that the cosets of A agree with orbits of [L, R], that Q/M ∼= (Inn Q)/L1 and that one can define an abelian group on Q/N ×L1. We also explain why the study of finite conjugacy closed loops can be restricted to the case of N/A nilpotent. Group [L,R] is shown to be a subgroup of a power of A (which is abelian), and we prove that Q/N can be embedded into Aut ([L, R]). Finally, we describe all conjugacy closed loops of order pq.
STRUCTURAL INTERACTIONS OF CONJUGACY CLOSED LOOPS
"... Abstract. We study conjugacy closed loops by means of their multiplication groups. Let Q be a conjugacy closed loop, N its nucleus, A the associator subloop, and L and R the left and right multiplication groups, respectively. Put M = {a ∈ Q; La ∈R}. WeprovethatthecosetsofAagree with orbits of [L, R] ..."
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Abstract. We study conjugacy closed loops by means of their multiplication groups. Let Q be a conjugacy closed loop, N its nucleus, A the associator subloop, and L and R the left and right multiplication groups, respectively. Put M = {a ∈ Q; La ∈R}. WeprovethatthecosetsofAagree with orbits of [L, R], that Q/M ∼ = (Inn Q)/L1 and that one can define an abelian group on Q/N × Mlt1. We also explain why the study of finite conjugacy closed loops can be restricted to the case of N/A nilpotent. Group [L, R] isshowntobea subgroup of a power of A (which is abelian), and we prove that Q/N can be embedded into Aut([L, R]). Finally, we describe all conjugacy closed loops of order pq. Conjugacy closed loops have been defined independently by Soikis [15] and by Goodaire and Robinson [7], [8]. More recent results concerning their structure have been obtained by Basarab [1], Kunen [10], Drápal [4] and by Kinyon, Kunen and Phillips [11]. This paper can be regarded as a sequel to [4] since its main concern rests in
COMMUTATOR THEORY FOR LOOPS
"... Abstract. Using the FreeseMcKenzie commutator theory for congruence modular varieties as the starting point, we develop commutator theory for the variety of loops. The fundamental theorem of congruence commutators for loops relates generators of the congruence commutator to generators of the total ..."
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Abstract. Using the FreeseMcKenzie commutator theory for congruence modular varieties as the starting point, we develop commutator theory for the variety of loops. The fundamental theorem of congruence commutators for loops relates generators of the congruence commutator to generators of the total inner mapping group. We specialize the fundamental theorem into several varieties of loops, and also discuss the commutator of two normal subloops. Consequently, we argue that some standard definitions of loop theory, such as elementwise commutators and associators, should be revised and linked more closely to inner mappings. Using the new definitions, we prove several natural properties of loops that could not be so elegantly stated with the standard definitions of loop theory. For instance, we show that the subloop generated by the new associators defined here is automatically normal. We conclude with a preliminary discussion of abelianess and solvability in loops. 1.