Fitting subgroup

Origin
The notion of Fitting subgroup was introduced by Hans Fitting.

Symbol-free definition
The Fitting subgroup of a group is defined as the subgroup generated by all its nilpotent normal subgroups.

For a finite group, it can be defined in the following equivalent ways:


 * 1) It is the unique largest nilpotent characteristic subgroup, and hence, also the join of all nilpotent characteristic subgroups.
 * 2) It is the unique largest nilpotent normal subgroup, and hence, also the join of all nilpotent normal subgroups.
 * 3) it is the unique largest nilpotent subnormal subgroup, and hence, also the join of all nilpotent subnormal subgroups.

Definition with symbols
The Fitting subgroup of a group $$G$$, denoted as $$F(G)$$, is defined as the subgroup generated by all nilpotent normal subgroups of $$G$$. In the particular case where $$G$$ is finite, the Fitting subgroup is itself a nilpotent group and can be defined in the following equivalent ways:


 * 1) $$F(G)$$ is the unique largest nilpotent characteristic subgroup of $$G$$, and hence, also the join of all nilpotent characteristic subgroups of $$G$$.
 * 2) $$F(G)$$ is the unique largest nilpotent normal subgroup of $$G$$, and hence, also the join of all nilpotent normal subgroups of $$G$$.
 * 3) $$F(G)$$ is the unique largest nilpotent subnormal subgroup of $$G$$, and hence, also the join of all nilpotent subnormal subgroups of $$G$$.

For a finite group
For a finite group, the Fitting subgroup is the direct product of $$p$$-cores for all the primes $$p$$.

Larger subgroup-defining functions

 * Generalized Fitting subgroup is the product of the Fitting subgroup and the layer (the commuting product of components).

Smaller subgroup-defining functions

 * Frattini subgroup is contained inside the Fitting subgroup, for a finite group. This follows from the fact that the Frattini subgroup of a finite group is nilpotent.

Group properties satisfied
The Fitting subgroup of any group is a Fitting group, viz a group generated by normal nilpotent subgroups. For a finite group, the Fitting subgroup is a nilpotent group.

Subgroup properties satisfied

 * Characteristic subgroup: The Fitting subgroup is always characteristic.
 * Strictly characteristic subgroup: The Fitting subgroup is always strictly characteristic (also called distinguished): any surjective endomorphism of the whole group sends the Fitting subgroup to within itself.
 * Self-centralizing subgroup when the whole group is solvable.

Free operator
A group whose Fitting subgroup is trivial is termed a Fitting-free group. A group is Fitting-free if and only if it has no proper nontrivial normal Abelian subgroups.

Fixed-point operator
A group is its own Fitting subgroup if and only if it is a Fitting group. For finite groups, this is equivalent to the condition of being a nilpotent group.

Subgroup-defining function properties
The Fitting subgroup of the Fitting subgroup is the Fitting subgroup. The fixed points are precisely the Fitting groups.

Associated constructions
The ascending series wherein each successive quotient is the Fitting subgroup of the quotient by the lower term of the whole group, is termed the Fitting series. For finite groups, the Fitting series terminates in finitely many steps at the whole group if and only if the group is solvable.

The length of the Fitting series for a given group is termed its Fitting length.