Evolving groups

TL;DR

This paper introduces and studies evolving groups, revealing their structure as Sylow Tower groups and as semidirect products of nilpotent groups with specific automorphism actions, connecting them to Galois cohomology.

Contribution

It defines the class of evolving groups, proves their Sylow Tower property, and characterizes them as semidirect products with conjugation automorphisms, linking group theory to Galois cohomology.

Findings

Evolving groups are Sylow Tower groups.

Evolving groups are characterized as semidirect products of nilpotent groups.

Automorphisms in evolving groups map subgroups to conjugates.

Abstract

The class of evolving groups is defined and investigated, as well as their connections to examples in the field of Galois cohomology. Evolving groups are proved to be Sylow Tower groups in a rather strong sense. In addition, evolving groups are characterized as semidirect products of two nilpotent groups of coprime orders where the action of one on the other is via automorphisms that map each subgroup to a conjugate.