#
A Classical Group of Neutrosophic Triplet Groups Using {Z_{2p}, ×}

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## Abstract

**:**

## 1. Introduction

## 2. Basic Concepts

**Definition**

**1.**

## 3. The Classical Group of Neutrosophic Triplet Groups of $\mathbf{\{}{\mathit{Z}}_{\mathbf{2}\mathit{p}}\mathbf{,}\mathbf{\times}\mathbf{\}}$ and Its Properties

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

- 1.
- If $a\in {Z}_{2p}$ has $neut\left(a\right)$ and $anti\left(a\right)$, then a is even.
- 2.
- The only nontrivial neutral element is $p+1$ for all a, which contributes to neutrosophic triplet groups in G.

**Proof.**

**Definition**

**2.**

**Example**

**4.**

**Definition**

**3.**

**Example**

**5.**

**Example**

**6.**

**Example**

**7.**

**Theorem**

**3.**

- 1.
- $K=\{2,4,\dots ,2p-2\}\subseteq {Z}_{2p}$ has a pseudo primitive element $x\in K$ with ${x}^{p-1}=p+1$, where $p+1$ is the multiplicative identity of K.
- 2.
- K is a cyclic group under × of order $p-1$ generated by that x, and $p+1$ is the identity element of K.
- 3.
- S is a Smarandache semigroup.

**Proof.**

**Theorem**

**4.**

**Proof.**

## 4. Semi-Neutrosophic Triplets and Their Properties

**Example**

**8.**

**Definition**

**4.**

**Proposition**

**1.**

**Proof.**

**Example**

**9.**

**Definition**

**5.**

## 5. Discussions and Conclusions

- How many pseudo primitive elements are there in $\{{Z}_{2p},\times \}$, where p is an odd prime?
- Can $\{{Z}_{n},\times \}$, where n is any composite number different from $2p$, have pseudo primitive elements? If so, which idempotent serves as the identity?

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## Abbreviations

SVNS | Single Valued Neutrosophic Set |

DVNS | Double Valued Neutrosophic Set |

TRINS | Triple Refined Indeterminate Neutrosophic Set |

IFS | Intuitionistic Fuzzy Set |

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**MDPI and ACS Style**

W.B., V.K.; Kandasamy, I.; Smarandache, F.
A Classical Group of Neutrosophic Triplet Groups Using {*Z*_{2p}, ×}. *Symmetry* **2018**, *10*, 194.
https://doi.org/10.3390/sym10060194

**AMA Style**

W.B. VK, Kandasamy I, Smarandache F.
A Classical Group of Neutrosophic Triplet Groups Using {*Z*_{2p}, ×}. *Symmetry*. 2018; 10(6):194.
https://doi.org/10.3390/sym10060194

**Chicago/Turabian Style**

W.B., Vasantha Kandasamy, Ilanthenral Kandasamy, and Florentin Smarandache.
2018. "A Classical Group of Neutrosophic Triplet Groups Using {*Z*_{2p}, ×}" *Symmetry* 10, no. 6: 194.
https://doi.org/10.3390/sym10060194