Fitting subgroup

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

History

Origin

The notion of Fitting subgroup was introduced by Hans Fitting.

Definition

Symbol-free definition

The Fitting subgroup of a group is defined as the subgroup generated by all its nilpotent normal subgroups. When the group is finite, it can also be defined as the unique largest normal nilpotent subgroup.

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.

For a finite group

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

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

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.

In terms of the group property core operator

This property is obtained by applying the [[defining ingredient::group property core operator]][[obtained by applying::group property core operator| ]] to the property: [[defining ingredient::nilpotent group]]
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Subgroup properties satisfied

• The Fitting subgroup is a characteristic subgroup
• In fact, it is an intermediately characteristic subgroup, viz it is characteristic in all intermediate subgroups as well.

Effect of operators

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

Idempotence

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.

Intermediacy-preservation

This subgroup-defining function is intermediacy-preserved, in other words, the image of any intermediate subgroup between the group and the image of the function, is also the image

The Fitting subgroup of a group is also the Fitting subgroup of any intermediate group. This follows from the fact that it arises via application of the group property core operator.