## Involutory decomposition of groups into twisted subgroups and subgroups (2000)

Venue: | J. Group Theory |

Citations: | 7 - 2 self |

### BibTeX

@ARTICLE{Foguel00involutorydecomposition,

author = {Tuval Foguel and Abraham A. Ungar},

title = {Involutory decomposition of groups into twisted subgroups and subgroups},

journal = {J. Group Theory},

year = {2000},

volume = {3},

pages = {27--46}

}

### OpenURL

### Abstract

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

### Citations

251 | The computational structure of monotone monadic SNP and constraint satisfaction: A study through Datalog and group theory
- Feder, Vardi
- 1999
(Show Context)
Citation Context ...sted subgroups are subsets of groups, introduced by Aschbacher [1], which under general conditions are near subgroups. The concept of near subgroup of a finite group was introduced by Feder and Vardi =-=[6]-=- asatool to study problems in computational complexity involving the class NP. 1s2T. FOGUEL AND A.A. UNGAR In a previous article [7] we have shown that every gyrogroup is a twisted subgroup in some sp... |

199 |
A Course in the Theory of Groups
- Robinson
- 1995
(Show Context)
Citation Context ...class 2 is equivalent to the condition that (K, ⊙) is a group, as desired. � A 2-Engel group E is a group satisfying [[a, b],b] = 1. Engel groups are useful in various studies of nilpotency; see e.g. =-=[14]-=-. It becomes evident from the following Theorem that these are useful in the study of gyrogroups as well. Theorem 3.7. Let (K, ⊙) be the left gyrogroup associated with a group (K, ·). Then (K, ⊙) is a... |

136 | A Survey of Binary Systems - Bruck - 1971 |

111 | An introduction to the theory of groups - Rotman - 1994 |

57 |
An Introduction to MAGMA
- Cannon, Playoust
- 1993
(Show Context)
Citation Context ...rk 3.10. 4. Examples. Example 4.1. The lowest order of non-group gyrogroups generated from nilpotent groups of class 3 which are not of class 2 is 16. Using the software package MAGMA and its library =-=[4]-=- we found three nonisomorphic nilpotent groups of order 16 which are of class 3 but are not of class 2 (see Theorem 3.6). They generate three non-gyrocommutative gyrogroups of order 16, denoted by K16... |

45 | Maximal subgroups of finite groups
- Aschbacher, Scott
- 1985
(Show Context)
Citation Context ...ts that (i) any gyrotransversal groupoid is a left gyrogroup, and (ii) any left gyrogroup is a twisted subgroup in a specified group. Twisted subgroups are subsets of groups, introduced by Aschbacher =-=[1]-=-, which under general conditions are near subgroups. The concept of near subgroup of a finite group was introduced by Feder and Vardi [6] asatool to study problems in computational complexity involvin... |

17 | Thomas precession: its underlying gyrogroup axioms and their use in hyperbolic geometry and relativistic - Ungar - 1997 |

13 | Thomas precession and its associated grouplike structure - Ungar - 1991 |

8 | Thomas rotation and the parametrization of the Lorentz transformation group - Ungar - 1988 |

8 | The holomorphic automorphism group of the complex disk - Ungar - 1994 |

7 | On multiplication groups of loops - Niemenmaa, Kepka - 1990 |

7 | On the equivalence of categories of loops and homogeneous spaces - Sabinin - 1972 |

7 | Weakly associative groups - Ungar - 1990 |

5 | The relativistic noncommutative nonassociative group of velocities and the Thomas rotation - Ungar - 1989 |

5 | From Pythagoras to Einstein: the hyperbolic Pythagorean theorem - Ungar - 1998 |

4 | J.D.: Connected transversals to subnormal subgroups - Kepka, Phillips - 1997 |

4 | T.: On connected transversals to abelian subgroups in finite groups - Niemenmaa, Kepka - 1992 |

4 | Axiomatic approach to the nonassociative group of relativistic velocities, Found.Phys.Lett.2(1989),199–203 - Ungar |

3 |
precession: its underlying gyrogroup axioms and their use in hyperbolic geometry and relativistic
- Thomas
- 1997
(Show Context)
Citation Context ...elements which are respectively denoted by 1P ,1H and 1G. We note that in any left gyrogroup (P, ⊙) the gyroautomorphisms gyr [p, 1P ] and gyr [1P ,p] are the identity automorphism of P , as shown in =-=[23]-=-.s4 T. FOGUEL AND A.A. UNGAR The left gyrogroup P , considered as a set, is the transversal of H in G = P ⋊� H by Definition 2.2. In order to show that (P, ⊙) is a gyrotransversal of H in G it remains... |

3 | A Survey of Binary Systems (Springer-Verlag - BrucK - 1966 |

3 | On the structure of the inner mapping groups of loops - Niemenmaa |

2 | On connected transversals in the projective special linear group PSL(2,7 - Niemenmaa, Vesanen - 1994 |

1 | associative groups - Weakly - 1990 |

1 |
precession and its associated grouplike structure
- Thomas
- 1991
(Show Context)
Citation Context ...f the Einstein 2-dimensional gyrogroup (ℜ 2 c, ⊕E) are all rotations of the Euclidean plane ℜ 2 about its origin, but there is no gyroautomorphism that rotates the plane about its origin by π radians =-=[21]-=-. K16 contains a group H which is a normal subgroup of the gyrogroup K16 (see Definitions 4.7 and 4.8 in [7]). The quotient gyrogroup K16/H turns out to be an abelian group. Hence, we have in hand an ... |

1 | Pythagoras to Einstein: The hyperbolic Pythagorean theorem, Found - From - 1998 |