The profinite completion of certain residually-finitep-groups

It is proved that the profinite completions of the residually finitep-groups constructed by Gupta and Sidki are not torsion groups.This research was carried out at Erlangen with the assistance of an Alexander von Humboldt Fellowship.     

Reversible hypergroups

Using the properties of the characters of a finite group as a prototype, a new notion of a reversible hypergroupH is introduced which takes account of degrees and multiplicities. Many properties of finite abelian groups have analogues for reversible hypergroups. For example, the hypergroup algebra CH is defined by analogy with a group algebra. Using the representations of this algebra, reversible hypergroups are shown to be in one-to-one correspondence with abstract «character tables» (of which Brauer’s «pseudo groups» are examples). These possess the usual orthogonality relations, whether or not they are character tables of groups. Examples are given of reversible hypergroups which are not isomorphic with the set of characters of any finite group.     

Locally toroidal regular polytopes of rank 4

The paper studies various relationships between locally toroidal regular 4-polytopes of types {6, 3,p} and {3, 6, 3}. These relationships are based on corresponding relationships between the regular honeycombs with the same Schläfli-symbol in hyperbolic 3-space. Also the paper discusses regular tessellations (secretions of rank 3) which are locally inscribed into regular 4-polytopes. In particular, this leads to local criteria for the finiteness of the polytopes.