# Chernikov group

A Chernikov group is a group $G$ that has a normal subgroup $N$ such that $G/N$ is finite and $N$ is a direct product of finitely many quasicyclic groups.

The significance of this somewhat arbitrary-looking definition is that all such groups satisfy the minimal condition, and for a long time they were the only known groups with this property.

Chernikov groups are named after http://www-groups.dcs.st-and.ac.uk/ history/Biographies/Chernikov.htmlSergei Chernikov, who proved that every solvable group that satisfies the minimal condition is a Chernikov group. We can state this result in the form of the following theorem.

###### Theorem.

The following are equivalent for a group $G$:

• $G$ is a Chernikov group.

• $G$ is virtually abelian and satisfies the minimal condition.

• $G$ is virtually solvable and satisfies the minimal condition.

Title Chernikov group ChernikovGroup 2013-03-22 15:48:21 2013-03-22 15:48:21 yark (2760) yark (2760) 8 yark (2760) Definition msc 20F50 Černikov group MinimalCondition