This monograph is devoted to the study of Polygroup Theory. It begins with some basic results concerning group theory and algebraic hyperstructures, which represent the most general algebraic context, in which reality can be modeled. Most results on polygroups are collected in this book. Moreover, this monograph is the first book on this theory. The volume is highly recommended to theoreticians in pure and applied mathematics.

Preface v

1. A Brief Excursion into Group Theory 1

1.1 Introduction 1

1.2 The abstract definition of a group and some examples 3

1.3 Subgroups 7

1.4 Normal subgroups and quotient groups 11

1.5 Group homomorphisms 13

1.6 Permutation groups 18

1.7 Direct product 22

1.8 Solvable and nilpotent groups 24

2. Hypergroups 33

2.1 Introduction and historical development of hypergroups . 33

2.2 Definition and examples of hypergroups 38

2.3 Some kinds of subhypergroups 40

2.4 Homomorphisms of hypergroups 43

2.5 Regular and strongly regular relations 52

2.6 Complete hypergroups 60

2.7 Join spaces 73

3. Polygroups 83

3.1 Definition and examples of polygroups 83

3.2 Extension of polygroups by polygroups 86

3.3 Subpolygroups and quotient polygroups 89

3.4 Isomorphism theorems of polygroups 95

3.5 γ∗relation on polygroups 98

3.6 Generalized permutations 105

3.7 Permutation polygroups 107

3.8 Representation of polygroups 112

3.9 Polygroup hyperrings 119

3.10 Solvable polygroups 123

3.11 Nilpotent polygroups 131

4. Weak Polygroups 139

4.1 Weak hyperstructures 139

4.2 Weak polygroups as a generalization of polygroups 148

4.3 Fundamental relations on weak polygroups 153

4.4 Small weak polygroups 156

5. Combinatorial Aspects of Polygroups 163

5.1 Chromatic polygroups 163

5.2 Polygroups derived from cogroups 175

5.3 Conjugation lattice 180

Bibliography 189

Index 197

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