Frobenius group

Suppose a finite group contains a subgroup satisfying specific properties. Using that information, what can be said about the structure of itself? One way to tackle such a problem is via character theory (cf. also Character of a group), another is by viewing as a permutation group. A classical and beautiful application of character theory is provided in elucidating the structure of Frobenius groups. Namely, let . Assume that whenever . Then is a so-called Frobenius complement in ; the group is then a Frobenius group by definition. It was proved by G. Frobenius in 1901, see [a3], that the set

is in fact a normal subgroup of . Almost a century later, Frobenius' proof that is a subgroup of is still the only existing proof; it uses character theory! The normal subgroup is called the Frobenius kernel of .

It can be shown that , that and that the orders of and are relatively prime. Therefore, by the Schur–Zassenhaus theorem, all Frobenius complements in are conjugate to each other. Below, let be an element of a group and let be a subset of that group; let denote the set .

A finite Frobenius group with Frobenius complement and corresponding Frobenius kernel satisfies:

1) for all ;

2) for all ;

3) for all ;

4) every is conjugate to an element of ;

5) if , then is conjugate to every element of the coset ;

6) each non-principal complex irreducible character of induces irreducibly to .

As a converse, assume that some finite group contains a normal subgroup and some subgroup satisfying and . Then the statements 1)–6) are all equivalent to each other, and if one of them is true, then is a Frobenius complement of , turning into a Frobenius group with as corresponding Frobenius kernel. Even more general, if some finite group with proper normal subgroup satisfies 1), then, applying one of the Sylow theorems, it is not hard to see that all orders of and are relatively prime. Whence there exists a subgroup of satisfying and (by the Schur–Zassenhaus theorem). Thus, again is a Frobenius group with Frobenius complement and Frobenius kernel .

Viewed another way, suppose a finite group , containing a non-trivial proper subgroup , acts transitively on a finite set with , such that for some prescribed element and such that only the identity of leaves invariant more than one element of . Then is a Frobenius group with Frobenius complement . Any element of the Frobenius kernel acts fixed-point freely on , i.e. for each .

There is a characterization of finite Frobenius groups in terms of group characters only. Namely, let be a subgroup of a finite group satisfying . Then the following assertions are equivalent:

a) statement 6) above;

b) is a normal subgroup of and is a Frobenius group with Frobenius kernel . The step from b) to a) was known to Frobenius; the converse step with, in addition, normal in is surely due to Frobenius; however, the step from a) to b) with not necessarily normal in is due to E.B. Kuisch (see [a7]).

This characterization led Kuisch, and later R.W. van der Waall, to the study of so-called -modular Frobenius groups; see [a8]. Namely, let be a field of positive characteristic . Then is a -modular Frobenius group if it contains a non-trivial normal subgroup such that is a splitting field for the group algebra and if one of the following (equivalent) statements holds:

A) every non-principal irreducible -module has the property that the induced -module is irreducible;

B) for every -regular non-trivial element . Any featuring in A)–B) is a -modular Frobenius kernel.

In 1959, J.G. Thompson [a9] showed that for a "classical" Frobenius group , the Frobenius kernel is nilpotent (cf. also Nilpotent group), thereby solving a long-standing conjecture of W.S. Burnside. It was proved by H. Zassenhaus in 1939, [a10], that a Sylow -subgroup (cf. also Sylow subgroup) of a Frobenius complement of is cyclic (cf. also Cyclic group) when is odd, and cyclic or generalized quaternion if . He also proved that if is not solvable (cf. also Solvable group), then it admits precisely one non-Abelian composition factor, namely the alternating group on five symbols.

The situation is more involved for -modular Frobenius groups. Namely, a -modular Frobenius kernel is either solvable (cf. also Solvable group) or else and any non–Abelian composition factor of is isomorphic to for some integer .

Furthermore, assume that is not a -group. Then:

any Sylow -subgroup of is cyclic whenever is relatively prime to ;

any Sylow -subgroup of is cyclic or generalized quaternion if is odd. On the other hand, any non-trivial finite -group ( a prime number) is isomorphic to some quotient group , where is a suitable -modular Frobenius group with -modular Frobenius kernel . See also [a8].

Historically, finite Frobenius groups have played a major role in many areas of group theory, notably in the analysis of -transitive groups and finite simple groups (cf. also Transitive group; Simple finite group).

Frobenius groups can be defined for infinite groups as well. Those groups are the non-regular transitive permutation groups in which only the identity has more than one fixed point. Again, let consist of the identity and those elements of the Frobenius group not occurring in any point stabilizer (cf. also Stabilizer). Contrary to the finite case, it is now not always true that is a subgroup of . See [a2] for examples.

References

[a1]	Yu.G. Berkovich, E.M. Zhmud, "Characters of finite groups" , Amer. Math. Soc. (1998/9)
[a2]	J.D. Dixon, B. Mortimer, "Permutation groups" , GTM , 163 , Springer (1996)
[a3]	G. Frobenius, "Ueber auflösbare Gruppen IV" Sitzungsber. Preuss. Akad. Wissenschaft. (1901) pp. 1216–1230
[a4]	B. Huppert, "Endliche Gruppen" , I , Springer (1967)
[a5]	B. Huppert, "Character theory of finite groups" , Experim. Math. , 25 , de Gruyter (1998)
[a6]	I.M. Isaacs, "Character theory of finite groups" , Acad. Press (1976)
[a7]	E.B. Kuisch, R.W. van der Waall, "Homogeneous character induction" J. Algebra , 156 (1993) pp. 395–406
[a8]	E.B. Kuisch, R.W. van der Waall, "Modular Frobenius groups" Manuscripta Math. , 90 (1996) pp. 403–427
[a9]	J.G. Thompson, "Finite groups with fixed point free automorphisms of prime order" Proc. Nat. Acad. Sci. USA , 45 (1959) pp. 578–581
[a10]	H. Zassenhaus, "Ueber endliche Fastkörper" Abh. Math. Sem. Univ. Hamburg , 11 (1936) pp. 187–220