The notable exclusions from the family of automatic groups are those nilpotent groups which are not virtually abelian, and the fundamental groups of compact 3-manifolds based on the Nil or Sol geometries. Of these, the 3-manifold groups have been shown by Bridson and Gilman to lie in a family of groups defined by conditions slightly more general than those of automatic groups, that is, to have ...

Let [Formula: see text] be a finite group and fixed prime divisor of text]. We prove that if every maximal subgroup is nilpotent, or normal, has text]-order, then (1) solvable; (2) Sylow tower; (3) there exists at most one such neither text]-nilpotent nor text]-closed.

In this work we study some topics in logic and group theory, and naturally, their intersection. One important theme in this work is the notion of random groups, which is informally the study of “what properties do ‘most’ groups satisfy?” In Chapter 2, we are interested in the theory of a random group, i.e., the properties we are interested here are first-order properties. Knight conjectured tha...

An automorphism of a group G is called regular if it moves every element of G except the identity. BURNSIDE proved that a finite group G has a regular automorphism of order two if and only if G is an abelian group of odd order, and then the only such automorphism maps every element onto its inverse ([21, p. 230). More recently several authors considered the question: what groups can admit regul...

Let G be a finite group and α be an automorphism of G of order p n for an odd prime p. Suppose that α acts fixed point freely on every α-invariant p-section of G, and acts trivially or exceptionally on every elementary abelian α-invariant p-section of G. It is proved that G is a solvable p-nilpotent group of nilpotent length at most n + 1, and this bound is best possible.

We describe a nilpotent quotient algorithm for a certain type of infinite presentations: the so-called finite L-presentations. We then exhibit finite L-presentations for various interesting groups and report on the application of our nilpotent quotient algorithm to them. As result, we obtain conjectural descriptions of the lower central series structure of various interesting groups including t...

We associate with each graph (S, E) a 2-step simply connected nilpotent Lie group N and a lattice Γ in N . We determine the group of Lie automorphisms of N and apply the result to describe a necessary and sufficient condition, in terms of the graph, for the compact nilmanifold N/Γ to admit an Anosov automorphism. Using the criterion we obtain new examples of compact nilmanifolds admitting Anoso...