Involutory decomposition of groups into twisted subgroups and subgroups

Foguel, Tuval; Ungar, Abraham A.

doi:10.1515/jgth.2000.003

Cited by 61 publications

(55 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…As a byproduct we also find from the product decomposition (3.2) that the element h ∈ Inn(K) ⊂ G determined by any two elements D(k 1 ) and D(k 2 ) of the transversal D, Definition 2.8 of [7], is…”

Section: Left Gyrogroups As Gyrotransversals Of Groupsmentioning

confidence: 92%

“…Thus, for instance, the gyroautomorphisms of the Einstein 2-dimensional gyrogroup ( 2 c , ⊕ E ) are all rotations of the Euclidean plane 2 about its origin, but there is no gyroautomorphism that rotates the plane about its origin by π radians [21]. K 16 contains a group H which is a normal subgroup of the gyrogroup K 16 (see Definitions 4.7 and 4.8 in [7]). The quotient gyrogroup K 16 /H turns out to be an abelian group.…”

Section: Examplesmentioning

confidence: 99%

“…In a previous article [7] we have shown that every gyrogroup is a twisted subgroup in some specified group, and introduced the involutory decomposition of a group G = BH into a twisted subgroup B and subgroup H, demonstrating that the resulting twisted subgroups are gyrocommutative gyrogroups. Gyrogroups are grouplike objects which share analogies with groups.…”

Section: Introductionmentioning

confidence: 98%

“…A gyrogroup is a grouplike structure that is defined in [7] along with a weaker structure called a left gyrogroup. We show that any given group can be turned into a left gyrogroup which is, in turn, a gyrogroup if and only if the given group is central by a 2-Engel group.…”

Section: Introductionmentioning

confidence: 99%

“…Following the involutory decomposition that we presented in [7], we present in this article a non-involutory decomposition of groups into twisted subgroups and subgroups. The resulting twisted subgroups, in this case, turn out to be gyrogroups that need not be gyrocommutative.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Gyrogroups and the decomposition of groups into twisted subgroups and subgroups

Foguel¹,

Ungar²

2001

Pacific J. Math.

Self Cite

View full text Add to dashboard Cite

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 and groups implies the existence of a general theory which underlies the theories of groups and gyrogroups and unifies them with respect to their central features. Accordingly, our goal is to construct finite and infinite gyrogroups, both gyrocommutative and non-gyrocommutaive, in order to demonstrate that gyrogroups abound in group theory of which they form an integral part.

show abstract

Section: Left Gyrogroups As Gyrotransversals Of Groupsmentioning

confidence: 92%

Section: Examplesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Gyrogroups and the decomposition of groups into twisted subgroups and subgroups

Foguel¹,

Ungar²

2001

Pacific J. Math.

Self Cite

View full text Add to dashboard Cite

show abstract

Placing the Hyperbolic Geometry of Bolyai and Lobachevsky Centrally in Special Relativity Theory: An Idea Whose Time has Returned

Ungar

Non-Euclidean Geometries

View full text Add to dashboard Cite

Möbius Transformation and Einstein Velocity Addition in the Hyperbolic Geometry of Bolyai and Lobachevsky

Ungar

2012

Springer Optimization and Its Applications

View full text Add to dashboard Cite

In this chapter, dedicated to the 60th Anniversary of Themistocles M. Rassias, Möbius transformation and Einstein velocity addition meet in the hyperbolic geometry of Bolyai and Lobachevsky. It turns out that Möbius addition that is extracted from Möbius transformation of the complex disc and Einstein addition from his special theory of relativity are isomorphic in the sense of gyrovector spaces.(1) Möbius addition in the ball R n c forms the algebraic setting for the Cartesian-Poincaré ball model of hyperbolic geometry, and (2) Einstein addition in the ball R n c forms the algebraic setting for the Cartesian-Beltrami-Klein ball model of hyperbolic geometry, just as the common (3) vector addition in the space R n forms the algebraic setting for the standard Cartesian model of Euclidean geometry.Remarkably, Items (1)-( 3) enable Möbius addition in R n c , Einstein addition in R n c , and the standard vector addition in R n to be studied comparatively, as in [72]. for all a, b, u, v ∈ R n c . Hence, gyr[u, v] is an isometry of R n c , keeping the norm of elements of the ball R n c invariant, (34) gyr[u, v]w = w Accordingly, gyr[u, v] represents a rotation of the ball R n c about its origin for any u, v∈R n c . The invertible self-map gyr[u, v] of R n c respects Einstein addition in R n c , (35) gyr[u, v](a⊕b) = gyr[u, v]a⊕gyr[u, v]bfor all a, b, u, v∈R n c , so that gyr [u, v] is an automorphism of the Einstein groupoid (R n c , ⊕). We recall that an automorphism of a groupoid (R n c , ⊕) is a bijective self-map of the groupoid R n c that respects its binary operation, that is, it satisfies (35). Under bijection composition the automorphisms of a groupoid (R n c , ⊕) form a group known as the automorphism group, and denoted Aut(R n c , ⊕). Being special automorphisms, gyrations gyr [u, v]∈Aut(R n c , ⊕), u, v ∈R n c , are also called gyroautomorphisms, gyr being the gyroautomorphism generator called the gyrator.

show abstract

Involutory decomposition of groups into twisted subgroups and subgroups

Cited by 61 publications

References 10 publications

Gyrogroups and the decomposition of groups into twisted subgroups and subgroups

Gyrogroups and the decomposition of groups into twisted subgroups and subgroups

Placing the Hyperbolic Geometry of Bolyai and Lobachevsky Centrally in Special Relativity Theory: An Idea Whose Time has Returned

Möbius Transformation and Einstein Velocity Addition in the Hyperbolic Geometry of Bolyai and Lobachevsky

Contact Info

Product

Resources

About