Book contents
- Frontmatter
- Contents
- Introduction
- Photograph
- 1 Automorphisms of solvable groups, Part I
- 2 Automorphisms of solvable groups, Part II
- 3 A survey of groups with a single defining relation
- 4 Some algorithms for computing with finite permutation groups
- 5 Five lectures on group rings
- 6 Buildings and group amalgamations
- 7 Finite presentability of S-arithmetic groups
- 8 Efficient presentations of GL(2, ℤ) and PGL(2, ℤ)
- 9 The commutator map
- 10 Polynomial functions and representations
- 11 On questions of Brauer and Feit
- 12 The Picard group and the modular group
- 13 Factor groups of the lower central series of free products of finitely generated abelian groups
- 14 Lattice ordered groups - a very biased survey
- 15 Totally orthogonal finite groups
- 16 One-relator products of groups
- 17 The Cavicchioli groups are pairwise non-isomorphic
- 18 Congruence and non-congruence subgroups of the modular group: a survey
- 19 Small cancellation theory with non-homogeneous geometrical conditions and application to certain Artin groups
- 20 The Lie algebra associated to the lower central series of a group
- 21 Algebraically closed locally finite groups
- 22 On power-commutative and commutation transitive groups
- 23 Dimension function for discrete groups
- 24 Coset graphs
- 25 Nilpotent quotient algorithms
- 26 Generators of p-groups
- 27 On the matrix groups associated to the isometries of the hyperbolic plane
- 28 A characteristic subgroup of N-stable groups
- 29 The isomorphism problem for integral group rings of finite nilpotent groups
- 30 Embedding the root group geometry of 2F4(q)
- 31 On generalized Frobenius complements
- 32 Subgroups of finite index in soluble groups: I
- 33 Subgroups of finite index in soluble groups: II
- 34 Some interconnections between group theory and logic
- 35 Groups covered by abelian subgroups
- 36 Embeddings of infinite permutation groups
- 37 Maximal subgroups of sporadic groups
26 - Generators of p-groups
Published online by Cambridge University Press: 05 March 2012
- Frontmatter
- Contents
- Introduction
- Photograph
- 1 Automorphisms of solvable groups, Part I
- 2 Automorphisms of solvable groups, Part II
- 3 A survey of groups with a single defining relation
- 4 Some algorithms for computing with finite permutation groups
- 5 Five lectures on group rings
- 6 Buildings and group amalgamations
- 7 Finite presentability of S-arithmetic groups
- 8 Efficient presentations of GL(2, ℤ) and PGL(2, ℤ)
- 9 The commutator map
- 10 Polynomial functions and representations
- 11 On questions of Brauer and Feit
- 12 The Picard group and the modular group
- 13 Factor groups of the lower central series of free products of finitely generated abelian groups
- 14 Lattice ordered groups - a very biased survey
- 15 Totally orthogonal finite groups
- 16 One-relator products of groups
- 17 The Cavicchioli groups are pairwise non-isomorphic
- 18 Congruence and non-congruence subgroups of the modular group: a survey
- 19 Small cancellation theory with non-homogeneous geometrical conditions and application to certain Artin groups
- 20 The Lie algebra associated to the lower central series of a group
- 21 Algebraically closed locally finite groups
- 22 On power-commutative and commutation transitive groups
- 23 Dimension function for discrete groups
- 24 Coset graphs
- 25 Nilpotent quotient algorithms
- 26 Generators of p-groups
- 27 On the matrix groups associated to the isometries of the hyperbolic plane
- 28 A characteristic subgroup of N-stable groups
- 29 The isomorphism problem for integral group rings of finite nilpotent groups
- 30 Embedding the root group geometry of 2F4(q)
- 31 On generalized Frobenius complements
- 32 Subgroups of finite index in soluble groups: I
- 33 Subgroups of finite index in soluble groups: II
- 34 Some interconnections between group theory and logic
- 35 Groups covered by abelian subgroups
- 36 Embeddings of infinite permutation groups
- 37 Maximal subgroups of sporadic groups
Summary
Since I started to work in the field of finite p-groups I have encountered, sometimes to my surprise, results concerned with the number of generators, and gradually I became convinced that this is an area worth investigating in its own right, with applications to other areas (such as padic Lie groups or Schur multipliers). In the present paper I try to collect some of the results about generators that seem to me to be the most interesting, admitting a natural bias towards my own work. This being a survey article, proofs are not usually given,except when not available elsewhere or as an illustration.
Some notation. The word “group” usually means a finite p-group, cl G and exp G are the class and exponent of G, d(G) and r(G) are the minimal numbers of generators and relations of G, G', Gi, ∏i(G), ф(G), Z(G), M(G) are the commutator subgroup, the i-th term of the lower central series, the subgroup generated by p -th powers, the Frattini subgroup,the centre and the Schur multiplier, Cn is a cyclic group of order n, wr stands for wreath product. Also, x is the smallest integer not less than the real number x, and logarithms are always to the base.
NUMBER OF GENERATORS
The simplest restriction is, of course, just to assume that dG) is given (or bounded). This is a very weak assumption. Indeed, any p-group can be embedded in a 2-generator one [NN]. Still, we mention two deep results. The first is Kostrikin's, stating that there are only finitely many p-groups of exponent p with a given number of generators.
- Type
- Chapter
- Information
- Proceedings of Groups - St. Andrews 1985 , pp. 273 - 281Publisher: Cambridge University PressPrint publication year: 1987