ENUMERATION OF FINITE GROUPS
CONTENTS I ELEM ENT A R Y R E S U L T S SOME BASIC OBSERVATIONS II GROUP S O F P R IME P O WE R O R D E R PRELIMINARIES TENSOR PRODUCTS AND EXTERIOR SQUARES OF ABELIAN GROUPS COMMUTATORS AND NILPOTENT GROUPS THE FRATTINI SUBGROUP LINEAR ALGEBRA ENUMERATING P-GROUPS: A LOWER BOUND RELATIVELY FREE GROUPS PROOF OF THE LOWER BOUND ENUMERATING P-GROUPS: UPPER BOUNDS AN ELEMENTARY UPPER BOUND AN OVERVIEW OF THE SIMS APPROACH ‘LINEARISING’ THE PROBLEM A SMALL SET OF RELATIONS PROOF OF THE UPPER BOUND III P Y BE R ’ S T H E O R E M SOME MORE PRELIMINARIES HALL SUBGROUPS AND SYLOW SYSTEMS THE FITTING SUBGROUP PERMUTATIONS AND PRIMITIVITY GROUP EXTENSIONS AND COHOMOLOGY GROUP EXTENSIONS COHOMOLOGY RESTRICTION AND TRANSFER THE MCIVER AND NEUMANN BOUND SOME REPRESENTATION THEORY SEMISIMPLE ALGEBRAS CLIFFORD’S THEOREM THE SKOLEM–NOETHER THEOREM EVERY FINITE SKEW FIELD IS A FIELD PRIMITIVE SOLUBLE LINEAR GROUPS SOME BASIC STRUCTURE THEORY THE SUBGROUP B THE ORDERS OF GROUPS CONJUGACY CLASSES OF MAXIMAL SOLUBLE SUBGROUPS OF SYMMETRIC GROUPS ENUMERATION OF FINITE GROUPS WITH ABELIAN SYLOW SUBGROUPS 12 1 COUNTING SOLUBLE A-GROUPS: AN OVERVIEW 12 2 SOLUBLE A-SUBGROUPS OF THE GENERAL LINEAR GROUP AND THE SYMMETRIC GROUPS 12 3 MAXIMAL SOLUBLE P?-A-SUBGROUPS 12 4 ENUMERATION OF SOLUBLE A-GROUPS MAXIMAL SOLUBLE LINEAR GROUPS 13 1 THE FIELD K AND A SUBFIELD OF K 13 2 THE QUOTIENT G/C AND THE ALGEBRA C 13 3 THE QUOTIENT B/A 13 4 THE SUBGROUP B 13 5 STRUCTURE OF G DETERMINED BY B CONJUGACY CLASSES OF MAXIMAL SOLUBLE SUBGROUPS OF THE GENERAL LINEAR GROUPS PYBER’S THEOREM: THE SOLUBLE CASE 15 1 EXTENSIONS AND SOLUBLE SUBGROUPS 15 2 PYBER’S THEOREM PYBER’S THEOREM: THE GENERAL CASE 16 1 THREE THEOREMS ON GROUP GENERATION 16 2 UNIVERSAL CENTRAL EXTENSIONS AND COVERING GROUPS 16 3 THE GENERALISED FITTING SUBGROUP 16 4 THE GENERAL CASE OF PYBER’S THEOREM IV OTHER TOP IC S ENUMERATION WITHIN VARIETIES OF ABELIAN GROUPS 17 1 VARIETIES OF ABELIAN GROUPS 17 2 ENUMERATING PARTITIONS 17 3 FURTHER RESULTS ON ABELIAN GROUPS ENUMERATION WITHIN SMALL VARIETIES OF A-GROUPS 18 1 A MINIMAL VARIETY OF A-GROUPS 18 2 THE JOIN OF MINIMAL VARIETIES ENUMERATION WITHIN SMALL VARIETIES OF P-GROUPS 19 1 ENUMERATING TWO SMALL VARIETIES 19 2 THE RATIO OF TWO ENUMERATION FUNCTIONS MISCELLANEA 20 1 ENUMERATING D-GENERATOR GROUPS 20 2 GROUPS WITH FEW NON-ABELIAN COMPOSITION FACTORS 20 3 ENUMERATING GRADED LIE RINGS 20 4 GROUPS OF NILPOTENCY CLASS SURVEY OF OTHER RESULTS 21 1 GRAHAM HIGMAN’S PORC CONJECTURE 21 2 ISOCLINISM CLASSES OF P-GROUPS 21 3 GROUPS OF SQUARE-FREE ORDER 21 4 GROUPS OF CUBE-FREE ORDER 21 5 GROUPS OF ARITHMETICALLY SMALL ORDERS 21 6 SURJECTIVITY OF THE ENUMERATION FUNCTION 21 7 DENSITIES OF CERTAIN SETS OF GROUP ORDERS 21 8 ENUMERATING PERFECT GROUPS SOME OPEN PROBLEMS REFERENCES