Chernikov group

A Chernikov group is a group $G$ that has a normal subgroup $N$ such that $G/N$ is finiteand $N$ 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 conditionis 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.

单词	ChernikovGroup
释义	Chernikov group A Chernikov group is a group $G$ that has a normal subgroup $N$ such that $G/N$ is finiteand $N$ 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 conditionis 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.
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