Some Neutrosophic Triplet Subgroup Properties and Homomorphism Theorems in Singular Weak Commutative Neutrosophic Extended Triplet Group

Abstract

In 2018, the study of neutrosophic triplet cosets and neutrosophic triplet quotient group of a neutrosophic extended triplet group was initiated with a follow up of the establishment of fundamental homomorphism
theorems for neutrosophic extended triplet group. But some lapses in these earlier results were identified and
revised through the introduction of special kind of weak commutative neutrosophic extended triplet group
(WCNETG) called perfect neutrosophic extended triplet group. Furthermore, neutro-homomorphism basic
theorem has been established for commutative neutrosophic extended triplet group. In this current work, the
generalization and extention of the above results was done by investigating neutro-homomorphism in singular
WCNETG. This was achieved with the introduction and study of some new types of NT-subgroups that are
right (left) cancellative, semi-strong, and maximally normal in a singular WCNETG. For any given non-empty
subset S and NT-subgroup H of a singular WCNETG X, some of these new NT-subgroups were shown to exist
as non-empty neutrosophic triplet normalizer, generated subset and centralizer of S, closure of H, derived subset
of X and center of X. With these, the first, second and third neutro-isomorphism and neutro-correspondence
theorems were established. This finally led to the proof of the neutro-Zassenhaus Lemma (Neutro-Butterfly
Theorem).