Semisimple algebraic group

This article defines a property that can be evaluated for an algebraic group. it is probably not a property that can directly be evaluated, or make sense, for an abstract group|View other properties of algebraic groups

Definition

Definition with symbols

An algebraic group $G$ over a field $k$ is said to be semisimple if it satisfies the following equivalent conditions:

1. It has no nontrivial normal solvable connected closed subgroup.
2. Its radical is the trivial subgroup.

Relation with other properties

Stronger properties

Stronger properties of algebraic groups include:

• Simple algebraic group

Weaker properties

Weaker properties of algebraic groups include:

• Reductive algebraic group