Fitting chain

Fitting series, nilpotent series

A standard way to decompose a finite solvable group into nilpotent sections (cf. also Nilpotent group). The Fitting chains achieve this with shortest possible length. This shortest possible length, the length of a Fitting series, is called the Fitting length of the group. The ascending Fitting chain (or upper nilpotent series) starts at the identity subgroup and builds up, each time with the largest subgroup of the group which contains the previous one and whose quotient by the previous one is nilpotent. The descending Fitting chain (or lower nilpotent series) starts with the group itself and builds successive smallest normal subgroups (cf. Normal subgroup) such that the quotient of the previous subgroup by the new subgroup is nilpotent. In the case of solvable groups, both series terminate, one with the whole group and the other with the identity.

Let $G$ be a finite group. Let $\mathcal F(G)$ denote the Fitting subgroup of $G$, and let $G^\mathcal N$ denote the smallest normal subgroup of $G$ such that $G/G^\mathcal N$ is nilpotent. The upper Fitting series is

$$F_0=\mathcal F_0(G)\leq F_1=\mathcal F_1(G)\leq\dotsb\leq F_n=\mathcal F_n(G)\leq\dotsb,$$

where $F_0=1$ and $F_i=\mathcal F(G/F_{i-1})$ for $i=1,2,\dotsc$. The lower Fitting series is

$$H_0\geq H_1\geq H_2\geq\dotsb\geq H_n\geq\dotsb,$$

where $H_0=G$ and $H_i=(H_{i-1})^\mathcal N$ for $i=1,2,\dotsc$. Each term in each of these series is a characteristic subgroup of $G$.

These characteristic subgroups are basic in describing the solvable group $G$. The ascending Fitting factors $\mathcal F_i(G)/\mathcal F_{i-1}(G)$ are nilpotent subgroups, and they give important information about the structure of $G$. Analogues of these concepts can also be obtained by replacing throughout the term "nilpotent" by some suitable other collection of groups, such as a Fitting class. Further details can be found in [a1], [a2], [a3].

References

[a1]	K. Doerk, T. Hawkes, "Finite soluble groups" , de Gruyter (1992)
[a2]	B. Huppert, "Endliche Gruppen I" , Springer (1967)
[a3]	B. Huppert, N. Blackburn, "Finite groups II" , Springer (1982)