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The Fascination of Groups by F.J. Budden

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This book is the best introduction to group theory I have ever read. It serves as an excellent resource for anyone interested in the mathematics of symmetry and its applications in art and design. Written in a clear and engaging style, it assumes no prior knowledge of group theory. The book is filled with illustrations and examples that effectively clarify the concepts being discussed. While most books on group theory become difficult to read and understand for non-mathematicians after a few chapters, this one stands out—it is well-organized and easy to follow.

Internet archive has a copy of the book available for borrowing. You can find it here.

Contents

Preface xi

Notation xvii

  1. Mathematical structure

  2. Putting things together: binary operations on a set Laws of composition: illustrations of binary operations on numbers, on sets, on people, on matrices, on ordered pairs and triples, in geometry. Unary operations. Matrices: a brief summary. Commutative and non-commutative operations. Exercises.

  3. Self-contained systems: closure Binary operations within a set. Illustrations; counter-examples—failure for closure. Closure by including extra elements, or by omitting elements. Exercises.

  4. Combined Ops: composition of operations Composition of geometrical transformations, of functions by successive substitution, of permutations, of matrices. Games with operations. Notation—first operation on the right: juxtaposition—the multiplicative notation. Matrices describing symmetries; permutation matrices. Successive transformations of the points of a circle. Exercises.

  5. Associativity Punctuation in mathematics. Testing for associativity. Examples of operations which are associative, and counter-examples where associativity fails. Associativity of mappings: examples. Exercises.

  6. Status Quo: identity elements Examples of identity elements. Formal definition. Finite arithmetics. Cases when identity elements are obscure. Geometrical examples. Left and right identities. Exercises.

  7. As You Were! - Inverses The idea of inverses—inverse permutations and their dual aspects. Inverses in finite arithmetic and geometric examples. General definition. The operation Δ on sets. Solving linear equations and singular matrices. \((xy)^{-1} = y^{-1}x^{-1}\). Exercises.

  8. Group Structure Requirements of a group—definition and examples of finite and infinite groups. Structure tables for finite groups. A group of six permutations. Introduction to groups of symmetries, including the two-group and further symmetry groups. Abelian groups. Groups of two-dimensional transformations. Systems that fail to be groups. Exercises.

  9. Properties of Groups Cancellation laws. The Latin-square property and its proof. Solving equations within groups. Use of Latin-square properties to complete group tables. Cayley's theorem and its verification. Regular representation of finite groups using matrices. Algebra within groups with practical examples. Exercises.

  10. Period (Order) of an Element: Permutations and Cycles Definition and finding the period of an element using group tables. Subgroups generated by single elements. Period of permutations—including cycles and cyclic permutations. Period of disjoint cycles and overlapping cycles. Periods of functions and matrices, including those with complex terms. Infinite period and the infinite cyclic group. Exercises.

  11. Carbon Copy Groups: Abstract Groups and Isomorphism The Klein four-group in thirteen different situations. Abstract groups and isomorphism of finite groups. Identifying different groups of order 4 and order 6. Establishing isomorphisms and their definitions. Infinite group isomorphism. Isomorphisms between additive and multiplicative groups in finite arithmetic. Modular arithmetic slide rules. Automorphisms of \(C_4\), \(D_2\), and \(C_6\). Inner automorphisms and their applications. Exercises.

  12. Cyclic Groups Various realizations of the cyclic group of order 6. Recurring decimals and cyclic groups of prime order. Regular polygons and cyclic groups of composite order. \(n\)th roots of unity and the cyclotomic equation. Related problems. The infinite cyclic group. Exercises.

  13. The Dihedral Group Direct and opposite symmetries, including mirror reflections. The full group of the regular hexagon (\(D_6\)). Other realizations of \(D_6\), \(D_3\), and \(D_4\). Reflections in intersecting mirrors and kaleidoscope symmetries. The group \(D_3\) of the equilateral triangle, isomorphic to \(S_3\). Permutations of vertices—fixed and moving axes. Cayley's theorem illustrated with a triangular prism. Subgroups of \(D_6\). The general dihedral group of order \(2n\): generators and defining relations. The infinite dihedral group. Groups generated by elements of period 2. Exercises.

  14. Groups Within Groups: Subgroups Definition and examples, including Lagrange’s theorem. Cyclic subgroups generated by single elements. Center of a group. Subgroups of \(S_4\). Normalizers and centralizers for subgroup identification. Non-cyclic subgroups generated by multiple elements. Subgroups in infinite groups like \(C_4^+\), \(C_x\), and their transformations. Finite groups of plane transformations and their subgroups. Exercises.

  15. Group Specifications: Generators and Defining Relations Groups requiring multiple generators. Defining relations and their independence. Group of quaternions and dicyclic groups. Cayley diagrams, including the Cayley graph for \(Q_8\) (the quaternion group). Additional notes on Cayley diagrams. Exercises.

  16. Bigger and Better Groups: Direct Product Groups Definition and examples, including the construction of new groups like \(C_6 \times C_2\). Associativity and commutativity of direct products, including non-Abelian cases. Periods of elements in direct product groups. The "packing-case" group and groups of order 24. Exercises.

  17. Catalogue of Groups: Symmetry Groups Table of all groups of order \( n \leq 12 \). Symmetry groups of two- and three-dimensional figures. The rotation group; enantiomorphs and the full group. Group of the regular tetrahedron; representation by matrices. Cube and octahedron; representation by matrices and by permutations. Icosahedral group \( A_5 \); the full group \( A_5 \times C_2 \). Exercises.

  18. Permutations Odd and even permutations – inversions of order. Change of parity due to a single transposition. Permutations as a product of transpositions. Pairing odd and even permutations – alternating groups. A set of permutations to represent \( S_4 \); two generators only needed. Resolution of a permutation into disjoint cycles. The period of a permutation is the LCM of the lengths of its disjoint cycles. Manipulation of cycles; overlapping cycles. The group of a polynomial. Cross-ratio; the six cross-ratios of four numbers. Exercises.

  19. Cosets in Finite and Infinite Groups: Equivalence Classes Cosets in the group \( D_4 \). Reading of cosets from the group table. Interpretation of cosets; illustrations. Preview of normal subgroups. Properties of cosets. Cosets in infinite groups: groups of vectors; of real and complex numbers; of plane transformations. Lagrange’s theorem – proof. Construction of structure tables. Equivalence relations and equivalence classes; partitioning into disjoint subsets. Cosets as equivalence classes. A binary operation on subsets – the product set. Product of subsets from infinite and finite groups. Product of cosets. Exercises.

  20. Conjugate Elements: Normal Subgroups (1) The transform of an element. Conjugate elements have the same period. Similar permutations. Reflection in a moved axis; successive reflections in moved axes. Successive rotations about moved vertices. Transformed operations in general; further illustrations. Conjugacy classes; finding conjugacy classes – the ‘snap’ method. The transforming element. Conjugacy classes and cosets; centralizers. The transform of a given subgroup; conjugate subgroups. Normal subgroups; methods for determining whether a given subgroup is normal. Normalizer of a subset. Exercises.

  21. Homomorphism: Quotient Groups: Normal Subgroups (2) Homomorphic mappings. Many-to-one mappings in general. Homomorphisms of finite groups. Kernel of a homomorphism. Normal subgroups; groups of cosets; quotient groups. Homomorphic images of Abelian groups. Failure when a subgroup is not normal. Product of cosets of normal subgroups. Quotient (or factor) groups. Quotient groups in direct product groups. Chains of normal subgroups. Simple groups: insolubility of the quintic equation. Proof that \( A_5 \) is simple, and that \( A_5 \) is the only normal subgroup of \( S_5 \). Infinite groups with finite and infinite quotient groups. Summary. Exercises.

  22. Automorphisms Inner and outer automorphisms. Automorphisms of Abelian and non-Abelian groups. Automorphisms of \( S_5 \). Proof that inner automorphisms form a normal subgroup of the full automorphism group.

  23. Groups and Music Musical pitch; the octave; musical intervals. The harmonic series. The perfect fifth. The pentatonic scale; the Pythagorean scale; just intonation. Equal temperament; tempered intervals – the group \( C_{12} \). Groups and musical form – the round, the canon, and the fugue. Imitation, inversion, augmentation, sequences. Exercises.

  24. Ringing the Changes: Groups and Campanology The campanologist’s rules for ringing changes and the principles of composition of methods. Three and four bells. Methods for producing twenty-four changes on four bells using group theory. Five bells – Stedman Doubles; leads and plain courses, subgroups, and cosets. Six bells; bobs and singles; Plain Bob Minor. Symmetry; other methods.

  25. Groups in Geometrical Situations

    • Dihedral groups on a circle and on a parabola generated by two involutions.
    • Poncelet's porism.
    • Relettering of geometrical configurations.
    • The complete quadrilateral; the orthocentric quadrilateral and the nine-point circle.
    • Pappus' theorem; Pascal's hexagon; Desargues' perspective triangle theorem.
    • Stabilizer subgroups.

  26. Patterns

    • Patterns obtained by systematic repetition of a motif.
    • The point groups \(C_n\) and \(D_n\).
    • Analytical and synthetic approaches.
    • Fundamental regions.
    • The seven frieze patterns.
    • Classification of patterns.
    • The two-dimensional (wallpaper) patterns.
    • The twelve plane patterns that contain opposite isometries.
    • Sub-patterns; subgroups of plane patterns; normal subgroups.
    • Exercises.

Appendices Answers Bibliography Index

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