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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 10 (4 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
Involutory Decomposition of Groups Into Twisted Subgroups and Subgroups
"... . An involutory decomposition is a decomposition, due to an involution, of a group into a twisted subgroup and a subgroup. We study unexpected links between twisted subgroups and gyrogroups. Twisted subgroups arise in the study of problems in computational complexity. In contrast, gyrogroups are gro ..."
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Cited by 8 (6 self)
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. An involutory decomposition is a decomposition, due to an involution, of a group into a twisted subgroup and a subgroup. We study unexpected links between twisted subgroups and gyrogroups. Twisted subgroups arise in the study of problems in computational complexity. In contrast, gyrogroups are grouplike structures which first arose in the study of Einstein's velocity addition in the special theory of relativity. Particularly, we show that every gyrogroup is a twisted subgroup and that, under general specified conditions, twisted subgroups are gyrocommutative gyrogroups. Moreover, we show that gyrogroups abound in group theory and that they possess rich structure. x1. Introduction Under general conditions, twisted subgroups are near subgroups [1]. Feder and Vardi [4] introduced the concept of a near subgroup of a finite group as a tool to study problems in computational complexity involving the class NP . Aschbacher provided a conceptual base for studying near subgroups demonstrating...
Zavarnitsine, Lagrange’s Theorem for Moufang loops
, 2005
"... We prove that the order of any subloop of a finite Moufang loop is a factor of the order of the loop, thus obtaining an analog of Lagrange’s theorem for finite Moufang loops. 1. ..."
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We prove that the order of any subloop of a finite Moufang loop is a factor of the order of the loop, thus obtaining an analog of Lagrange’s theorem for finite Moufang loops. 1.
Sylow’s theorem for Moufang loops
, 709
"... For finite Moufang loops, we prove an analog of the first Sylow theorem giving a criterion of the existence of a pSylow subloop. We also find the maximal order of psubloops in the Moufang loops that do not possess pSylow subloops. Keywords: Moufang loop, Sylow’s theorem, group with triality MSC20 ..."
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For finite Moufang loops, we prove an analog of the first Sylow theorem giving a criterion of the existence of a pSylow subloop. We also find the maximal order of psubloops in the Moufang loops that do not possess pSylow subloops. Keywords: Moufang loop, Sylow’s theorem, group with triality MSC2000: 08A05, 20N05, 17D05