Semisimple group

The term semisimple has also been used at some places for a group whose solvable radical is trivial, which is equivalent to being a Fitting-free group

Symbol-free definition
A group is said to be semisimple if it occurs as a central product of (possibly more than two) quasisimple groups.

Definition with symbols
A group $$G$$ is said to be semisimple if there are subgroups $$G_1, G_2, \ldots, G_r$$ such that:


 * Each $$G_i$$ is quasisimple
 * The $$G_i$$s generate $$G$$
 * The group $$[G_i, G_j]$$ is trivial for all $$i \ne j$$

Metaproperties
Every finite group can be realized as a subgroup of a semisimple group. This follows from the fact that every finite group can be realized as a subgroup of a simple non-Abelian group.

Every quotient of a semisimple group is semisimple. This follows from the fact that every quotient of a quasisimple group is quasisimple, and that a central product is preserved on going to the quotient.

A direct product of semisimple groups is semisimple. In fact, any central product of semisimple groups is semisimple.