This introduction to the theory of groups covers all topics likely to be encountered in undergraduate courses. The book takes as its theme the various fundamental classification theorems in finite group theory, and the text is further explained in numerous examples and exercises, and summaries at the end of each chapter. Definitions and examples 1(8) Maps and relations on sets 8(10) Elementary consequences of the definitions 18(12) Subgroups 30(8) Cosets and Lagrange's Theorem 38(11) Error-correcting codes 49(10) Normal subgroups and quotient groups 59(9) The Homomorphism Theorem 68(9) Permutations 77(12) The Orbit-Stabiliser Theorem 89(9) The Sylow Theorems 98(8) Applications of Sylow theory 106(6) Direct products 112(8) The classification of finite abelian groups 120(8) The Jordan--Holder Theorem 128(9) Composition factors and chief factors 137(9) Soluble groups 146(9) Examples of soluble groups 155(8) Semidirect products and wreath products 163(11) Extensions 174(9) Central and cyclic extensions 183(9) Groups with at most 31 elements 192(10) The projective special linear groups 202(11) The Mathieu groups 213(9) The classification of finite simple groups 222(12) The classical groups 223(7) The projective special linear groups 223(1) The unitary groups 223(3) The symplectic groups 226(2) The orthogonal groups 228(2) Groups of Lie type 230(1) The sporadic groups 231(3) A Prerequisites from number theory and linear algebra 234(4) A.1 Number theory 234(1) A.2 Linear algebra and determinants 235(3) B Groups of order < 32 238(5) C Solutions to exercises 243(32) Bibliography 275(2) Index 277