Chernikov group

A Chernikov group is a group G that has a normal subgroupMathworldPlanetmath N such that G/N is finite and N is a direct productMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath 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 history/Biographies/Chernikov.htmlSergei Chernikov, who proved that every solvable groupMathworldPlanetmath that satisfies the minimal condition is a Chernikov group. We can state this result in the form of the following theorem.


The following are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath 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
Canonical name ChernikovGroup
Date of creation 2013-03-22 15:48:21
Last modified on 2013-03-22 15:48:21
Owner yark (2760)
Last modified by yark (2760)
Numerical id 8
Author yark (2760)
Entry type Definition
Classification msc 20F50
Synonym Černikov group
Related topic MinimalCondition