# Fitting subgroup

The characteristic subgroup of a group generated by all the nilpotent normal subgroups of ; it is also called the Fitting radical. It was first studied by H. Fitting [1]. For finite groups, the Fitting subgroup is nilpotent and is the unique maximal nilpotent normal subgroup of the group. For a finite group the following relations hold:

where denotes the Frattini subgroup of , and is the commutator subgroup of .

#### References

[1] | H. Fitting, "Beiträge zur Theorie der Gruppen endlicher Ordnung" Jahresber. Deutsch. Math.-Verein , 48 (1938) pp. 77–141 |

[2] | A.G. Kurosh, "The theory of groups" , 1–2 , Chelsea (1955–1956) (Translated from Russian) |

[3] | D. Gorenstein, "Finite groups" , Harper & Row (1968) |

#### Comments

A finite group is called quasi-nilpotent if and only if for every chief factor of every automorphism of induced by an element of is inner. Let be the centre of . Inductively define , by the condition . The are all normal subgroups. The series is the ascending central series of . Let , the so-called hypercentre of . A group is semi-simple if and only if it is the direct product of non-Abelian simple groups. A group is quasi-nilpotent if and only if is semi-simple. The generalized Fitting subgroup of a finite group is the set of all elements of which induce an inner automorphism on every chief factor of . It is a characteristic subgroup of and contains every subnormal quasi-nilpotent subgroup of . This property can therefore also be used to define it.

Let be the lower central series of , i.e. , the commutator subgroup of with . Let be the generalized Fitting subgroup of ; then is a perfect quasi-nilpotent characteristic subgroup. It is sometimes called the layer of .

#### References

[a1] | B. Huppert, "Endliche Gruppen" , 1 , Springer (1967) |

[a2] | B. Huppert, "Finite groups" , 2–3 , Springer (1982) |

[a3] | D.J.S. Robinson, "A course in the theory of groups" , Springer (1982) |

**How to Cite This Entry:**

Fitting subgroup.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Fitting_subgroup&oldid=17227