Fitting subgroup

The characteristic subgroup $F(G)=F$ of a group $G$ generated by all the nilpotent normal subgroups of $G$; 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 $G$ the following relations hold:

$$ [F,F]\subseteq\Phi\subseteq F \quad and \quad F/\Phi=F(G/\Phi) , $$ where $\Phi$ denotes the Frattini subgroup of $G$, and $[F,F]$ is the commutator subgroup of $F$.

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 $G$ is called quasi-nilpotent if and only if for every chief factor $H$ of $G$ every automorphism of $H$ induced by an element of $G$ is inner. Let $Z(G)=Z_0(G)=\{g\in G:gh=hg\text{ for all }h\text{ such that }h\in G\}$ be the centre of $G$. Inductively define $Z_i(G)\supseteq Z_{i-1}(G)$, $i=1,2,\ldots$ by the condition $Z_i(G)/Z_{i-1}(G)=Z(G/Z_{i-1}(G))$. The $Z_i(G)$ are all normal subgroups. The series $Z_0(G)\subseteq Z_1(G)\subseteq\ldots$ is the ascending central series of $G$. Let $Z_\infty(G)=\bigcup_i Z_i(G)$, the so-called hypercentre of $G$. A group $G$ is semi-simple if and only if it is the direct product of non-Abelian simple groups. A group $G$ is quasi-nilpotent if and only if $G/Z_\infty(G)$ is semi-simple. The generalized Fitting subgroup of a finite group $G$ is the set of all elements $x$ of $G$ which induce an inner automorphism on every chief factor of $G$. It is a characteristic subgroup of $G$ and contains every subnormal quasi-nilpotent subgroup of $G$. This property can therefore also be used to define it.

Let $G=\gamma_1(G)\supseteq\gamma_2(G)\supseteq\ldots$ be the lower central series of $G$, i.e. $\gamma_n(G)=[G,\gamma_{n-1}(G)]$, the commutator subgroup of $G$ with $\gamma_{n-1}(G)$. Let $F^\ast(G)$ be the generalized Fitting subgroup of $G$; then $E(G)=\bigcap_{n\geq 1}\gamma_n(F^\ast(G))$ is a perfect quasi-nilpotent characteristic subgroup. It is sometimes called the layer of $G$.

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)