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A Note on Fuzzy Automorphism and Inner Automorphism of Groups

Shiv Narain, Sunil Kumar, Sandeep Kumar, Gaurav Mittal

Abstract

The fuzzification of classical set theory came into existence when Zadeh [1] laid down the concept of a fuzzy set as a generalization of a crisp set. The objective of this paper is to extend the concept of fuzzy endomorphism to fuzzy automorphism. Notions of fuzzy inner automorphism and fuzzy inner automorphism induced by a fuzzy subgroup are introduced. Finally, we obtain the fuzzy analogues of well-known results of classical group theory.

A Note on Fuzzy Automorphism and Inner Automorphism of Groups

Abstract

The fuzzification of classical set theory came into existence when Zadeh [1] laid down the concept of a fuzzy set as a generalization of a crisp set. The objective of this paper is to extend the concept of fuzzy endomorphism to fuzzy automorphism. Notions of fuzzy inner automorphism and fuzzy inner automorphism induced by a fuzzy subgroup are introduced. Finally, we obtain the fuzzy analogues of well-known results of classical group theory.

Paper Structure

This paper contains 5 sections, 24 theorems, 108 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. fuzzy automorphism and fuzzy inner automorphism of a group
  4. fuzzy permutation induced by a fuzzy subgroup and Cayley Theorem
  5. fuzzy inner automorphism of a group G induced by a fuzzy normal subgroup

Key Result

Theorem \oldthetheorem

Let $f:G\cdots \to G'$ be a fuzzy homomorphism. Then

Theorems & Definitions (56)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.1
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Theorem \oldthetheorem
  • ...and 46 more