Fitting length

nilpotent length

The Fitting length, also known as nilpotent length, of a finite solvable group (cf. also Finite group) provides a measure of how far the group is from being nilpotent. For any finite solvable group $G$, the ascending and the descending Fitting chains of $G$ both have the same number of distinct elements (cf. also Fitting chain). The length of the chain (that is, the number of distinct elements in either chain minus one) is called the Fitting length of $G$. Thus, the trivial group has Fitting length $0$, and any non-trivial nilpotent group has Fitting length $1$, whereas any finite solvable non-nilpotent group will have Fitting length at least $2$; see any standard reference such as [a2], [a3], [a4] for details.

As one measure of the complexity of a solvable group, the Fitting height is related to many other such measures. Thus, it is related to the number of distinct irreducible character degrees, to the derived length, to the number of elements needed to generate the Sylow subgroups of the group, to the derived length of the Sylow subgroups or to their nilpotent class. It is, however, its relationship with fixed points of automorphism groups that is the most striking.

Frobenius's conjecture, proved by J.G. Thompson [a9], states that if $G$ is any finite group admitting an automorphism $\phi$ such that $\phi$ has prime order and no element of $G$ except the identity is fixed by $\phi$, then $G$ is nilpotent. This can be extended to the following conjecture. Let $G$ be a finite group and let $A$ be a group of automorphisms of $G$ such that $( | A | , | G | ) = 1$ and no element of $G$ except the identity is fixed by all the elements of $A$. Then $G$ is solvable and the Fitting length of $G$ is bounded above by the length of the longest chain of subgroups in $A$. Denoting by $h ( G )$ the Fitting length of $G$ and by $\operatorname{l} ( A )$ the length of the longest chain of subgroups of $A$, the conjecture states that $h ( G ) \leq \text{l} ( A )$. This is known to be true in many cases [a6] and to be the best possible in all cases [a7]. Similar bounds can be obtained when the group of automorphism does have some fixed point. For example, [a8], if $G$ is a finite solvable group and $A$ is a solvable group of automorphisms of $G$ such that $( | A | , | G | ) = 1$, then $h ( G ) \leq h ( C _ { G } ( A ) ) + 2 \text{l} ( A )$, where $C _ { G } ( A )$ denotes the subgroup of the elements of $G$ that are fixed under every automorphism in $A$. In some cases one can also give bounds when $A$ is not solvable, see [a5], but these bounds are bigger.

Fixed-point-free automorphisms are closely related to Carter subgroups of solvable groups (cf. also Carter subgroup). In [a1], E.C. Dade proved that there is an exponential function $f$ such that if $G$ is a finite solvable group and $C$ is its Carter subgroup, then $h ( G ) \leq f ( \text{l} ( C ) )$, and conjectured that there actually exists a linear function $f$ with the same properties. In [a7], it is proved that there exits a quadratic function $f$ which satisfies the condition whenever $C$ is a direct product of elementary Abelian groups.

References

[a1]	E.C. Dade, "Carter subgroups and Fitting heights of finite solvable groups" Illinois J. Math. , 13 (1969) pp. 449–514
[a2]	K. Doerk, T. Hawkes, "Finite soluble groups" , de Gruyter (1992)
[a3]	B. Huppert, "Endliche Gruppen I" , Springer (1967)
[a4]	B. Huppert, N. Blackburn, "Finite Groups II" , Springer (1982)
[a5]	H. Kurzweil, "Auflösbare Gruppen auf denen nicht auflösbare Gruppen operieren" Manuscripta Math. , 41 (1983) pp. 233–305
[a6]	A. Turull, "Fixed point free action with some regular orbits" J. Algebra , 194 (1997) pp. 362–377
[a7]	A. Turull, "Character theory and length problems" , Finite and Locally Finite Groups (Istanbul, 1994) , Kluwer Acad. Publ. (1995) pp. 377–400
[a8]	A. Turull, "Fitting height of groups and of fixed points" J. Algebra , 86 (1984) pp. 555–566
[a9]	J.G. Thompson, "Finite groups with fixed-point-free automorphisms of prime order" Proc. Nat. Acad. Sci. USA , 45 (1959) pp. 578–581