We show that the commutator subgroup G ′ of a classical knot group G need not have subgroups of every finite index, but it will if G ′ has a surjective homomorphism to the integers and we give an exact criterion for that to happen. We also give an example of a smoothly knotted S n in S n+2 for all n ≥ 2 whose infinite cyclic cover is not simply connected but has no proper finite covers.

Let F be a field of characteristic different from 2. It is shown that the problems of classifying (i) local commutative associative algebras over F with zero cube radical, (ii) Lie algebras over F with central commutator subalgebra of dimension 3, and (iii) finite p-groups of exponent p with central commutator subgroup of order p3 are hopeless since each of them contains • the problem of classi...

This paper addresses the problems of estimating the normal covariance and precision matrices. A commutator subgroup of lower triangular matrices is considered for deriving a class of invariant estimators. The class shows inadmissibility of the best invariant and minimax estimator of the covariance matrix relative to quadratic loss. Also, in estimation of the precision matrix, a dominance result...

Let K be the kernel of an epimorphism G→ Z, where G is a finitely presented group. If K has infinitely many subgroups of index 2, 3 or 4, then it has uncountably many. Moreover, if K is the commutator subgroup of a classical knot group G, then any homomorphism from K onto the symmetric group S2 (resp. Z3) lifts to a homomorphism onto S3 (resp. alternating group A4).

Elements of the commutator subgroup of a free group F can be presented as values of canonical forms, called Wicks forms. We show that, starting from sufficiently high genus g, there is a sequence of words wg which can be presented by f .g/ distinct Wicks forms, where f .g/ > gŠ. Moreover we may choose these words wg to be square-free. Mathematics Subject Classification (2000). 20E05, 20F10.

The main aim of this paper is to show that every two-generator subgroup of any one-relator group with torsion is either a free product of cyclic groups or is a one-relator group with torsion. This result is proved by using techniques for reducing pairs of elements in certain HNN groups. These techniques not only apply to one-relator groups with torsion but also to a large number of other groups...

We shall term a group G supersoluble if every homomorphic image H9*l of G contains a cyclic normal subgroup different from 1. Supersoluble groups with maximum condition, in particular finite supersoluble groups, have been investigated by various authors: Hirsch, Ore, Zappa and more recently Huppert and Wielandt. In the present note we want to establish the close connection between supersoluble ...

Abstract A group is called quasihamiltonian if all its subgroups are permutable, and we say that a subgroup Q of G permutably embedded in $\langle Q,g\rangle $ for each element g . It proved here contains normal such $G/Q$ ?ernikov, then has finite index; moreover, periodic, it ?ernikov N $G/N$ quasihamiltonian. This result should be compared with theorems Schlette stating over centre, abelian-...