Introduction to Fp-Groups# Introduction Overview Definitions and Notation Construction of FP-Groups Relations Quotient Group Constructors Example: Defining S4 as a Quotient Group Example: Quotient Constructor without a Free Group FP-Group constructor Example: Binary Tetrahedral Group Example: Quotient of an FP-Group Example: Coxeter's Parametrised Family of Groups Presentation Operations Example: Simplifying the Fibonacci Group F(8) Operations on Words Presentations for Popular Groups Example: Abelian Group as an FP-Group Example: Symmetric Group Sym(8) as an FP-Group Constructions for FP-Groups Example: Direct and Free Products of FP-Groups Example: FP-Group Isomorphic to GL(3,4) Properties of an FP-group Example: Proving a Coxeter Group Infinite Example: Automatic Structure of the Infinite Dihedral Group Subgroups Subgroup Constructor Example: Subgroup of Index 448 Coset Enumeration Example: Coset Table of an Index 15 Subgroup Coset Action Example: Permutation Representation of G2(3) Enumeration of FI-Subgroups Example: Tutte's 8-Cage and the Conder Graph Example: Conjugacy Classes via Low Index Subgroups Operations for FI-Subgroups Example: Operations on a Subgroup of Index 448 Properties of Subgroups Example: Subgroup Predicates in a Space Group Presentations for Subgroups Example: Presentation of an Index 3 Subgroup Example: Rewriting a Presentation of L2(7) Finite FP-Groups Example: Order of a Finite FP-Group Example: Cayley Graph of a Finite FP-Group Homomorphisms of FP-Groups Homomorphism Constructor Example: Symmetric Group as Image of a Braid Group Searching for Homomorphisms Example: Homomorphism onto A5 Searching for Isomorphisms Example: Isomorphism Search for Hillman's Groups Example: Isomorphism Search for Neumann's Groups Quotient Groups Abelian Quotient Example: Structure of the Fibonacci Group F(7) Example: Abelian Quotient Invariants of a Subgroup \(p\)-Quotient Example: Largest 2-Quotient of Class 6 Example: 3-Quotient of a Group of Exponent 9 Example: Metabelian 5-Quotient Nilpotent Quotient Example: Class 2 Nilpotent Quotient Example: Free Nilpotent Groups via Nilpotent Quotient Example: Nilpotent Quotient of a Baumslag-Solitar Group Soluble Quotient Example: Soluble Quotient of a Two-Generator Group Example: Order of a Large Soluble Quotient Simple Group Quotients Example: Simple Quotients of a Perfect Group \({\rm L}_2\)-Quotients Example: L2-Quotients of FP-Groups Example: Groups with Infinitely Many L2-Quotients Example: Matrix Groups of an L2-Quotient Bibliography