A group G is called morphic if every endomorphism α:G→G for which Gα◃G satisfies G/Gα≅ker (α). Call an endomorphism α∈end(G) regular if αβα=α for some β∈end(G), and call α unit regular if β can be chosen to be an automorphism of G. The main purpose of this paper is to prove the following group-theoretic analogue of a theorem of Ehrlich: if G is a morphic group, an endomorphism α:G→G for which Gα◃G is unit regular if and only if it is regular. As an application, a cancellation theorem is proved that characterizes the morphic groups among those with regular endomorphism monoids.

In a soluble group G, if the metabelian subgroups have finite (or polycyclic, or Černikov) quotients by their centres, then so too does G.

Finite groups with abelian Sylow p-subgroups for certain primes p are characterized in terms of arithmetical properties of commutators.

A group G satisfies the second Engel condition [X,Y,Y ]=1 if and only if x commutes with xy, for all x,y∈G. This paper considers the generalization of this condition to groups G such that, for fixed positive integers r and s, xr commutes with (xs)y for all x,y∈G. Various general bounds are proved for the structure of groups in the corresponding variety, defined by the law [Xr,(Xs)Y]=1.

We study commutation properties of subsets of right-angled Artin groups and trace monoids. We show that if Γ is any graph not containing a four-cycle without chords, then the group G(Γ) does not contain four elements whose commutation graph is a four-cycle; a consequence is that G(Γ) does not have a subgroup isomorphic to a direct product of non-abelian groups. We also obtain corresponding and more general results in the monoid case.

We describe the group of all reflection-preserving automorphisms of an imprimitive complex reflection group. We also study some properties of this automorphism group.

The arithmetic is interpreted in all the groups of Richard Thompson and Graham Higman, as well as in other groups of piecewise affine permutations of an interval which generalize the groups of Thompson and Higman. In particular, the elementary theories of all these groups are undecidable. Moreover, Thompson's group F and some of its generalizations interpret the arithmetic without parameters.

In 2006 we completed the proof of a five-part conjecture that was made in 1977 about a family of groups related to trivalent graphs. This family covers all 2-generator, 2-relator groups where one relator specifies that a generator is an involution and the other relator has three syllables. Our proof relies upon detailed but general computations in the groups under question. The proof is theoretical, but based upon explicit proofs produced by machine for individual cases. Here we explain how we derived the general proofs from specific cases. The conjecture essentially addressed only the finite groups in the family. Here we extend the results to infinite groups, effectively determining when members of this family of finitely presented groups are simply isomorphic to a specific quotient.

This note contains some remarks on generating pairs for automorphism groups of free groups. There has been significant use of electronic assistance. Little of this is used to verify the results.

An enumeration result for orientably regular hypermaps of a given type with automorphism groups isomorphic to PSL(2,q) or PGL(2,q) can be extracted from a 1969 paper by Sah. We extend the investigation to orientable reflexible hypermaps and to nonorientable regular hypermaps, providing many more details about the associated computations and explicit generating sets for the associated groups.

We prove that a finitely generated semigroup whose word problem is a one-counter language has a linear growth function. This provides us with a very strong restriction on the structure of such a semigroup, which, in particular, yields an elementary proof of a result of Herbst, that a group with a one-counter word problem is virtually cyclic. We prove also that the word problem of a group is an intersection of finitely many one-counter languages if and only if the group is virtually abelian.

For a group G and a real number x≥1 we let sG(x) denote the number of indices ≤x of subgroups of G. We call the function sG the subgroup density of G, and initiate a study of its asymptotics and its relation to the algebraic structure of G. We also count indices ≤x of maximal subgroups of G, and relate it to symmetric and alternating quotients of G.

A star is a planar set of three lines through a common point in which the angle between each pair is 60∘.A set of lines through a point in which the angle between each pair of lines is 60 or 90∘ is star-closed if for every pair of its lines at 60∘ the set contains the third line of the star. In 1976 Cameron, Goethals, Seidel and Shult showed that the indecomposable star-closed sets in Euclidean space are the root systems of types An, Dn, E6, E7 and E8. This result was a key part of their determination of all graphs with least eigenvalue −2. Subsequently, Cvetković, Rowlinson and Simić determined all star-closed extensions of these line systems. We generalize this result on extensions of line systems to complex n-space equipped with a hermitian inner product. There is one further infinite family, and two exceptional types arising from Burkhardt and Mitchell’s complex reflection groups in dimensions five and six. The proof is a geometric version of Mitchell’s classification of complex reflection groups in dimensions greater than four.

An extension of a result of Sela shows that if Γ is a torsion-free word hyperbolic group, then the only homomorphisms Γ→Γ with finite-index image are the automorphisms. It follows from this result and properties of quasiregular mappings, that if M is a closed Riemannian n-manifold with negative sectional curvature (), then every quasiregular mapping f:M→M is a homeomorphism. In the constant-curvature case the dimension restriction is not necessary and Mostow rigidity implies that f is homotopic to an isometry. This is to be contrasted with the fact that every such manifold admits a non-homeomorphic light open self-mapping. We present similar results for more general quotients of hyperbolic space and quasiregular mappings between them. For instance, we establish that besides covering projections there are no π1-injective proper quasiregular mappings f:M→N between hyperbolic 3-manifolds M and N with non-elementary fundamental group.

An element in a free group is a proper power if and only if it is a proper power in every nilpotent factor group. Moreover there is an algorithm to decide if an element in a finitely generated torsion-free nilpotent group is a proper power.

This paper concerns parabolic submonoids of a class of monoids known as singular Artin monoids. The latter class includes the singular braid monoid— a geometric extension of the braid group, which was created for the sole purpose of studying Vassiliev invariants in knot theory. However, those monoids may also be construed (and indeed, are defined) as a formal extension of Artin groups which, in turn, naturally generalise braid groups. It is the case, by van der Lek and Paris, that standard parabolic subgroups of Artin groups are canonically isomorphic to Artin groups. This naturally invites us to consider whether the same holds for parabolic submonoids of singular Artin monoids. We show that it is in fact true when the corresponding Coxeter matrix is of ‘type FC’ hence generalising Corran's result in the ‘finite type’ case.

Palindromes are those reduced words of free products of groups that coincide with their reverse words. We prove that a free product of groups G has infinite palindromic width, provided that G is not the free product of two cyclic groups of order two (Theorem 2.4). This means that there is no uniform bound k such that every element of G is a product of at most k palindromes. Earlier, the similar fact was established for non-abelian free groups. The proof of Theorem 2.4 makes use of the ideas by Rhemtulla developed for the study of the widths of verbal subgroups of free products.

The aim of this work is to offer a new characterization of the Hilbert symbol Q*p from the commutator of a certain central extension of groups. We obtain a characterization for Q*p (p≠2) and a different one for Q*2.

We define a notion of conjugacy in singular Artin moniods, and solve the corresponding conjugacy problem for finite types. We sgiw that this definition is appropriate to describe type (1) singular Markov moves on singular braids. Parabolic submonoids of singular Artin monoids are defined and, in finite type, are shown to be singular Artin monoids. Solutions to conjugacy-type problems of parabolic submonoids are described. Geometric objects defined by Fenn, Rolfsen and Zhu, called (j, k)-bands, are algebraically characterised, and a procedure is given which determines when a word represents a (j, k)-band.

The conjugacy classes of so-called special involutions parameterize the constituents of the action of a finite Coxeter group on the cohomology of the complement of its complexified hyperplane arrangement. In this note we give a short intrinsic characterisation of special involutions in terms of so-called bulky parabolic subgroups.