This article defines a subgroup-defining function, viz., a rule that takes a group and outputs a unique subgroup
View a complete list of subgroup-defining functions OR View a complete list of quotient-defining functions
- 1 History
- 2 Definition
- 3 Relation with other subgroup-defining functions
- 4 Group properties satisfied
- 5 Subgroup properties satisfied
- 6 Effect of operators
- 7 Subgroup-defining function properties
- 8 Associated constructions
- 9 Computation
The notion of Fitting subgroup was introduced by Hans Fitting.
For a finite group, it can be defined in the following equivalent ways:
- It is the unique largest nilpotent characteristic subgroup, and hence, also the join of all nilpotent characteristic subgroups.
- It is the unique largest nilpotent normal subgroup, and hence, also the join of all nilpotent normal subgroups.
- 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 , denoted as , is defined as the subgroup generated by all nilpotent normal subgroups of . In the particular case where is finite, the Fitting subgroup is itself a nilpotent group and can be defined in the following equivalent ways:
- is the unique largest nilpotent characteristic subgroup of , and hence, also the join of all nilpotent characteristic subgroups of .
- is the unique largest nilpotent normal subgroup of , and hence, also the join of all nilpotent normal subgroups of .
- is the unique largest nilpotent subnormal subgroup of , and hence, also the join of all nilpotent subnormal subgroups of .
For a finite group
For a finite group, the Fitting subgroup is the direct product of -cores for all the primes .
Relation with other subgroup-defining functions
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
In terms of the group property core operator
This property is obtained by applying the group property core operator to the property: nilpotent group
View other properties obtained by applying the group property core operator
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. For full proof, refer: Solvable implies Fitting subgroup is self-centralizing
Effect of operators
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.
Subgroup-defining function properties
This subgroup-defining function is idempotent. In other words, applying this twice to a given group has the same effect as applying it once
The Fitting subgroup of the Fitting subgroup is the Fitting subgroup. The fixed points are precisely the Fitting groups.
Associated quotient-defining function
The quotient-defining function associated with this subgroup-defining function is: Fitting quotient
Associated ascending series
The associated ascending series to this subgroup-defining function is: Fitting series
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.
The command for computing this subgroup-defining function in Groups, Algorithms and Programming (GAP) is:FittingSubgroup
View other GAP-computable subgroup-defining functions