In a group G, um (G) denotes the subgroup of the elements which normalize every subnormal subgroup of G with defect at most m. The m-Wielandt series of G is then defined in a natural way. G is said to have finite m-Wielandt length if it coincides with a term of its m-Wielandt series. We investigate the structure of infinite groups with finite m-Wielandt length.

This note gives a necessary condition, in terms of graded actions, for an inverse semigroup to be a full amalgam. Under a mild additional hypothesis, the condition becomes sufficient.

One of the converse statements to Lagrange's theorem is that, for each subgroup H of G and any prime factor p of |G: H|, there exists a subgroup K such that H≤K≤G with |K: H | = p. This paper treats integers n such that all groups of order n have this property.

A group G is locally graded if every finitely generated nontrivial subgroup of G has a nontrivial finite image. Let N (2, k)* denote the class of groups in which every infinite subset contains a pair of elements that generate a nilpotent subgroup of class at most k. We show that if G is a finitely generated locally graded N (2, k)*-group, then there is a positive integer c depending only on k such that G/Zc (G) is finite.

We prove that there is no algorithm to determine when an amalgam of finite rings (or semigroups) can be embedded in the class of rings or in the class of finite rings (respectively, in the class of semigroups or in the class of finite semigroups). These results are in marked contrast with the corresponding problems for groups where every amalgam of finite groups can be embedded in a finite group.

We provide a wedge decomposition of the homotopy type of the p-subgroup complex in the case of a finite solvable group G. In particular, this includes a new proof of the result of Quillen which says that this complex is contractible if and only if there is a non-trivial normal p-subgroup in G. We also provide reduction formulas for the G-module structure of the homology groups. Our results are obtained with diagram-methods by gluing the p-subgroup complex of G along the p-subgroup complex of = G/N for a normal p′-subgroup of G.

Let N be a finitely generated normal subgroup of a finitely generated group G. We show that if the trivial subgroup is tame in the factor group G/N, then N is that in G. We also give a short new proof of the fact that quasiconvex subgroups of negatively curved groups are tame. The proof utilizes the concept of the geodesic core of the subgroup and is related to the Dehn algorithm.

The associativity of the regular semidirect product of existence varieties introduced by Jones and Trotter was proved under certain condition by Reilly and Zhang. Here we estabilsh associativity in many new cases. Moreover, we prove that the regular semidirect product is right distributive with respect to the join operation. In particular, both associativity and right distriutivity yiel within the varieties of completely simple semigroups. Analogous results are obtainedj for e-pseudovarieties of finite regular semigroups.

We show that diagrammatically reducible two-complexes are characterized by the property: every finity subconmplex of the universal cover collapses to a one-complex. We use this to show that a compact orientable three-manifold with nonempty boundary is Haken if and only if it has a diagrammatically reducible spine. We also formulate an nanlogue of diagrammatic reducibility for higher dimensional complexes. Like Haken three-manifolds, we observe that if n ≥ 4 and M is compact connected n-dimensional manifold with a traingulation, or a spine, satisfying this property, then the interior of the universal cover of M is homeomorphic to Euclidean n-space.

It is known that the direct product of two automatic groups is automatic. The notion of automaticity bas been extended to semigroups, and this for groups has been generalized to automatic monoids. However, the direct product of two automatic semigroups need not be finitely generated and hence not automatic.

Robertson, Ruškuc and Wiegold have determined necessary and sufficient conditions for the direct product of two finitely generated semigroups to be finitely generated. Building on this, we prove the following. Let S and T be automatic semigroups; if S and T are infinite, then S × T is automatic if and only if S2 = S and T2 = T; if S is finite and T is infinite, then S × T is automatic if and only if S2 = S. As a consequence, we have that, if S and T are automatic semigroups, then S × T is automatic if and only if S × T is finitely generated.

As a sequel to the previous two papers of the second author, we investigate the structure of medial idempotent groupoids by Pn-sequences. To complete the series of research, this paper has theree purposes. First, we summarize some results in the previous papers so that this paper can cover the materials presented there. Secondly, using earlier results, we prove a few theorems which show the importance of the medial law in controlling the growth of Pn-sequences of groupoids. Finally, we state some problems and conjectures raised during the series of research.

Let V be a finite dimensional Hermitian vector space and K be a compact Lie subgroup of U(V) for which th representation of K on C[V] is multiplicity free. One obtains a canonical basis {pα} for the space C [VR]k of K-invariant polynomials on VR and also a basis {q's. The polynomial pα's yields the homogeneous component of highest degree in qα. The coefficient that express the qα's in terms of the pβ's are the generalized binomial coeffficients of Yan. The main result in this paper shows tht these numbers are rational.

In this paper a necessary and sufficient condition will be given for groups to be ν-isologic, with respect to a given variety of groups ν. Its is also shown that every ν-isologism family of a group contains a ν-Hopfian group. Finally we show that if G is in the variety ν, then every ν-covering group of G is a Hopfian group.

The main result indicates that every finitely generated, residually finite, torsion-free, cohopfian group having on free Abelian subgroup of rank two is hyperhopfian. The argument relies on earlier work and ideas of Hirshon. As a corollary, fundamental groups of all closed hyperbolic manifolds are hyperhopfian.

A finite dinilpotent group G is one that can be written as the product of two finite nilpotent groups, A and B say. A finite dinilpotent group is always soluble. If A is abelian and B is metabelian, with |A| and|B| coprime, we show that a bound on the derived length given by Kazarin can be improved. We show that G has derived length at most 3 unless G contains a section with a well defined structure: in particular if G is of odd order, G has derived length at most 3.

We present some questions that we feel are important and interesting for the theory of finite p-groups, and survey known results regarding these questions.

Mike, the other year you asked me how I saw the future of p-group theory. All I can offer in answer are more questions.

A group A is said to be endoprimal if its term functions are precisely the functions which permute with all endomorphisms of A. In this paper we describe endoprimal groups in the following three classes of abelian groups: torsion groups, torsionfree groups of rank at most 2, direct sums of a torsion group and a torsionfree group of rank 1.

A group G belongs to the class W if G has non-nilpotent proper subgroups and is isomorphic to all of them. The main objects of study are the soluble groups in W that are not finitely generated. It is proved that there are no torsion-free groups of this sort, and a reasonable classification is given in the finite rank case.

Let K be an arbitrary field of characteristic 2, F a free group of countably infinite rank. We construct a finitely generated fully invariant subgroup U in F such that the relatively free group F/U satisfies the maximal condition on fully invariant subgroups but the group algebra K (F/U) does not satisfy the maximal condition on fully invariant ideals. This solves a problem posed by Plotkin and Vovsi. Using the developed techniques we also construct the first example of a non-finitely based (nilpotent of class 2)-by-(nilpotent of class 2) variety whose Abelian-by-(nilpotent of class at most 2) groups form a hereditarily finitely based subvariety.

The object of this paper is to study the sequence of torsion-free ranks of the quotients by the terms of the lower central series of a finitely generated group. This gives rise to the introduction into the study of finitely generated, residually torison-free nilpotent groups of notions relating to the Gelfand-Kirillov dimension. These notions are explored here. The main result concerning the sequences alluded to is the proof that there are continuously many such sequences.