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
22 - On power-commutative and commutation transitive 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
INTRODUCTION
A group G is a PC-group (power commutative) if for any elements a,b in G of infinite orders, proper powers of a,b permute only if a,b permute, i.e. [am,bn] = 1, m,n ≠ 0, implies [a,b] = 1. G is a CT-group (commutationtransitive) if for any a,b,c in G, b noncentral, [a,b] = [b,c] = 1 implies [a,c] = 1. In [1], B. Fine notes that free groups and Fuchsian groups are both PC and CT groups and poses the problem of finding further classes of PC and of CT groups (the latter problem is attributed to L. Greenberg). In this note we construct various examples of PC and CT groups and consider generalized free products and HNN-extensions of such groups.
GL(2,K)
The example of a free nilpotent group of class at least 4 shows that not all PC-groups are CT-groups. Below (Theorem 2) we give an example of a CT-group which is not a PC-group. On the other hand, a group with presentation (a,b,t : tat-1 = a-1), which is an HNN-extension of a free group is neither a CT-group nor a PC-group.
In this section we consider examples arising from linear groups. Let K be any field and GL(2,K) the general linear group of 2 x 2 matrices over K. If K contains elements of infinite orders GL(2,K) contains elements of infinite order which are non-central but have finite powers which are central, and, hence, GL(2,K) will not be a PC-group.
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- Proceedings of Groups - St. Andrews 1985 , pp. 249 - 253Publisher: Cambridge University PressPrint publication year: 1987
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