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Groupprops β

Fitting-free group

Definition

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 abelian normal subgroup.
  2. It has no nontrivial nilpotent normal subgroup.
  3. It has no nontrivial solvable normal subgroup.
  4. Its Fitting subgroup is trivial.
  5. Its solvable radical is trivial.

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

Equivalence of definitions

The key idea is to use the fact that any nontrivial solvable group has a nontrivial abelian characteristic subgroup. Further information: Equivalence of definitions of Fitting-free group

Relation with other properties

Stronger properties

Weaker properties