Proving group isomorphism theorems

How complex numbers get into play in a non-trivial way in real theories of gravitation is relevant since in a unified structure they should be able to relate in a natural way with quantum theories. For a long time this issue has been lingering on both relativistic formulations and quantum theories. We will analyze this fundamental subject under the light of new group isomorphism theorems linking local internal groups of transformations and local groups of spacetime transformations. The bridge between these two kinds of transformations is represented by new tetrads introduced previously. It is precisely through these local tetrad structures that we will provide a non-trivial answer to this old issue. These new tetrads have two fundamental building components, the skeletons and the gauge vectors. It is these constructive elements that provide the mathematical support that allows to prove group isomorphism theorems. In addition to this, we will prove a unique new property, the infinite tetrad nesting, alternating the nesting with non-Abelian tetrads in the construction of the tetrad gauge vectors. As an application we will demonstrate an alternative proof of a new group isomorphism theorem.

Download Full-text

On fundamental isomorphism theorems in soft subgroups

Soft Computing ◽

10.1007/s00500-020-05074-5 ◽

2020 ◽

Vol 24 (16) ◽

pp. 11841-11851

Author(s):

N. Çağman ◽

R. Barzegar ◽

S. B. Hosseini

Keyword(s):

Isomorphism Theorems

Download Full-text

Non-Lebesgue measurability of finite unions of Vitali selectors related to different groups

MATHEMATICA SCANDINAVICA ◽

10.7146/math.scand.a-128969 ◽

2021 ◽

Vol 127 (3) ◽

Author(s):

Venuste Nyagahakwa ◽

Gratien Haguma

Keyword(s):

Topological Group ◽

Finite Union ◽

Affine Transformations ◽

Real Numbers ◽

Group Isomorphism ◽

Lebesgue Measurability ◽

Dense Subgroups

In this paper, we prove that each topological group isomorphism of the additive topological group $(\mathbb{R},+)$ of real numbers onto itself preserves the non-Lebesgue measurability of Vitali selectors of $\mathbb{R}$. Inspired by Kharazishvili's results, we further prove that each finite union of Vitali selectors related to different countable dense subgroups of $(\mathbb{R}, +)$, is not measurable in the Lebesgue sense. From here, we produce a semigroup of sets, for which elements are not measurable in the Lebesgue sense. We finally show that the produced semigroup is invariant under the action of the group of all affine transformations of $\mathbb{R}$ onto itself.

Download Full-text

Fundamental Homomorphism Theorems for Neutrosophic Extended Triplet Groups

Symmetry ◽

10.3390/sym10080321 ◽

2018 ◽

Vol 10 (8) ◽

pp. 321 ◽

Cited By ~ 4

Author(s):

Mehmet Çelik ◽

Moges Shalla ◽

Necati Olgun

Keyword(s):

Group Theory ◽

Significant Role ◽

Classical Group ◽

Algebraic Structures ◽

Algebraic Systems ◽

Homomorphism Theorem ◽

Special Cases ◽

Isomorphism Theorems

In classical group theory, homomorphism and isomorphism are significant to study the relation between two algebraic systems. Through this article, we propose neutro-homomorphism and neutro-isomorphism for the neutrosophic extended triplet group (NETG) which plays a significant role in the theory of neutrosophic triplet algebraic structures. Then, we define neutro-monomorphism, neutro-epimorphism, and neutro-automorphism. We give and prove some theorems related to these structures. Furthermore, the Fundamental homomorphism theorem for the NETG is given and some special cases are discussed. First and second neutro-isomorphism theorems are stated. Finally, by applying homomorphism theorems to neutrosophic extended triplet algebraic structures, we have examined how closely different systems are related.

Download Full-text