Fitting-free group

Symbol-free definition
A group is said to be Fitting-free (or sometimes, semisimple) if it satisfies the following equivalent conditions:


 * 1) It has no nontrivial defining ingredient::abelian normal subgroup.
 * 2) It has no nontrivial defining ingredient::nilpotent normal subgroup.
 * 3) It has no nontrivial defining ingredient::solvable normal subgroup.
 * 4) Its defining ingredient::Fitting subgroup is trivial.
 * 5) Its defining ingredient::solvable radical is trivial.

When the group is finite, this is equivalent to the following:


 * It has no nontrivial defining ingredient::abelian characteristic subgroup
 * It has no nontrivial defining ingredient::nilpotent characteristic subgroup
 * It has no nontrivial defining ingredient::solvable characteristic subgroup
 * It has no nontrivial defining ingredient::elementary abelian normal subgroup
 * It has no nontrivial defining ingredient::elementary abelian characteristic subgroup

Equivalence of definitions
The key idea is to use the fact that any nontrivial solvable group has a nontrivial abelian characteristic subgroup.

Stronger properties

 * Simple non-Abelian group
 * Semisimple group

Weaker properties

 * Centerless group