A Chernikov group is a group $G$ that has a normal subgroup $N$ such that $G\over N$ is finite and $N$ is a direct product of finitely many quasicyclic groups.
PlanetMath
$(*)$ A periodic group of automorphisms of a Chernikov group is itself a Cernikov group.
What I have to prove is the following:
- $(*)$ implies that a periodic group of automorphisms of a Chernikov-by-cyclic group is also a Chernikov group.
Let $G$ be a Chernikov-by-cyclic group such that $AutG$ is periodic. Let $C\leq G$ the Chernikov subgroup by hypothesis such that $G\over C$ is cyclic. I have to show that $AutG$ is a Chernikov group.
I work out some infos:
- $AutC$ is a Chernikov group.
- $Aut{G\over C}$ is finite.
- $OutC$ is finite ("Finiteness Condition", vol. 1, Robinson)
- $AutC$ is finite (D. J. S. ROBINSON, Infinite soluble groups with no outer automorphisms, Rend.Sem. Mat. Univ. Padova, 62 (1980), 281-294.)
Clearly a finite group is a Chernikov group.
I miss how connect all this infos... Any ideas?