Vol 55, No 5 (2016)

The spectrum ω (G) of a finite group G is the set of orders of elements of G. Let S be a simple exceptional group of type E₆ or E₇. We describe all finite groups G such that S ≤ G ≤ Aut S and ω (G) = ω (S) and completes the study of the recognition-by-spectrum problem for all simple exceptional groups of Lie type.

Previously, the author defined the concept of a rigid (abstract) group. By analogy, a metabelian pro-p-group G is said to be rigid if it contains a normal series of the form G = G₁ ≥ G₂ ≥ G₃ = 1 such that the factor group A = G/G₂ is torsion-free Abelian, and G₂ being a Z_pA-module is torsion-free. An abstract rigid group can be completed and made divisible. Here we do something similar for finitely generated rigid metabelian pro-p-groups. In so doing, we need to exit the class of pro-p-groups, since even the completion of a torsion-free nontrivial Abelian pro-p-group is not a pro-p-group. In order to not complicate the situation, we do not complete a first factor, i.e., the group A. Indeed, A is simply structured: it is isomorphic to a direct sum of copies of Z_p. A second factor, i.e., the group G₂, is completed to a vector space over a field of fractions of a ring Z_pA, in which case the field and the space are endowed with suitable topologies. The main result is giving a description of coordinate groups of irreducible algebraic sets over such a partially divisible topological group.

Algebraic geometry over completely simple semigroups is considered. We prove theorems that define coordinate semigroups and irreducible coordinate semigroups in several classes of completely simple semigroups.