Fitting subgroup

The Fitting subgroup $$F(G)$$ of a finite group is the product of all normal nilpotent subgroups, equivalently the unique largest normal nilpotent subgroup. It gathers the normal local structure at every prime into one characteristic subgroup. In a finite solvable group it is also self-centralizing and provides the natural bottom layer for induction. ^{[@isaacs2008]}

Definition

Every normal nilpotent subgroup is the direct product of its Sylow subgroups, and this yields the canonical factorization

F(G)=\prod_{p\mid |G|}O_p(G).

Only primes for which $$O_p(G)>1$$ contribute nontrivial factors.

Fitting's product theorem

Fundamental properties

The Fitting subgroup has the following invariance and functorial properties:

$F(G)\operatorname{char}G$ , because every automorphism permutes the normal nilpotent subgroups;
if $N\triangleleft G$ , then $F(N)\leq F(G)$ ;
$F(G)N/N\leq F(G/N)$ , although equality need not hold;
$F(G\times H)=F(G)\times F(H)$ ;
$$F(G)=1$$ exactly when $$G$$ has no nontrivial normal $$p$$ -subgroup.

The last statement follows from the $$p$$ -core factorization. It does not say that $$G$$ has no nilpotent subgroups; it says that none of their nontrivial prime-power parts is normal in the whole group.

Self-centralization in solvable groups

Solvability cannot be omitted. If $$G$$ is nonabelian simple, then $$F(G)=1$$ while $$C_G(F(G))=G$$ . For arbitrary finite groups the correct replacement is the generalized Fitting subgroup

\[ F^*(G)=F(G)E(G), \]

where $$E(G)$$ is the central product of all components. It satisfies $C_G(F^*(G))\leq F^*(G)$ .

Minimal normal subgroups

If $$G$$ is solvable, every minimal normal subgroup $$N$$ is elementary abelian. Indeed, $$N'$$ is normal in $$G$$ , so minimality forces $$N'=1$$ ; applying the same argument to $$N^p$$ for a prime dividing $$|N|$$ makes $$N$$ an elementary abelian $$p$$ -group. Consequently

N\leq O_p(G)\leq F(G).

Every nontrivial finite solvable group therefore has $$F(G)>1$$ . This gives a standard induction scheme: choose a minimal normal subgroup inside $$F(G)$$ , then pass to $$G/N$$ .

The full complement statement is part of Gaschütz theory. In the elementary case, the Sylow factors of $$F(G)$$ can be treated as $\mathbf F_pG$ -modules, while the Frattini quotient removes nongenerators.

Examples

Fitting height

For a finite solvable group define recursively

F_0(G)=1, \qquad F_{i+1}(G)/F_i(G)=F\bigl(G/F_i(G)\bigr).

The least $$h$$ with $$F_h(G)=G$$ is the Fitting height. A nontrivial nilpotent group has height $$1$$ , while $$S_3$$ has height $$2$$ . The invariant counts how many nilpotent layers are needed to build a solvable group.

Applications

proving that finite solvable groups have nontrivial normal elementary abelian subgroups;
embedding $$G/F(G)$$ into automorphism groups of Fitting chief factors;
defining Fitting height and organizing induction on solvable groups;
adjoining the layer $$E(G)$$ to form $$F^*(G)$$ for nonsolvable groups;
controlling normal $$p$$ -structure in permutation and local group theory.

This article independently reorganizes the product theorem and its later subnormal consequences from Isaacs, with terminology cross-checked against Fitting's historical paper and the cited public encyclopedia revision. ^{[@isaacs2008]} ^{[@fitting1938]} ^{[@wikipedia-fitting]}