Frobenius group

Suppose a finite group $G$ contains a subgroup satisfying specific properties. Using that information, what can be said about the structure of $G$ itself? One way to tackle such a problem is via character theory (cf. also Character of a group), another is by viewing $G$ as a permutation group. A classical and beautiful application of character theory is provided in elucidating the structure of Frobenius groups. Namely, let $\{ 1 \} < H < G$. Assume that $H \cap g ^ { - 1 } H g = \{ 1 \}$ whenever $g \in G \backslash H$. Then $H$ is a so-called Frobenius complement in $G$; the group $G$ is then a Frobenius group by definition. It was proved by G. Frobenius in 1901, see [a3], that the set

\begin{equation*} N = \{ G \backslash ( \bigcup _ { x \in G } x ^ { - 1 } H x ) \} \bigcup \{ 1 \} \end{equation*}

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

It can be shown that $G = N H$, that $N \cap H = \{ 1 \}$ and that the orders of $N$ and $H$ are relatively prime. Therefore, by the Schur–Zassenhaus theorem, all Frobenius complements in $G$ are conjugate to each other. Below, let $t$ be an element of a group and let $S$ be a subset of that group; let $C _ { S } ( t )$ denote the set $\{ s \in S : s ^ { - 1 } t s = t \}$.

A finite Frobenius group $G$ with Frobenius complement $H$ and corresponding Frobenius kernel $N$ satisfies:

1) $C _ { G } ( n ) \leq N$ for all $1 \neq n \in N$;

2) $C _ { H } ( n ) = \{ 1 \}$ for all $1 \neq n \in N$;

3) $C _ { G } ( h ) \leq H$ for all $1 \neq h \in H$;

4) every $x \in G \backslash N$ is conjugate to an element of $H$;

5) if $1 \neq h \in H$, then $h$ is conjugate to every element of the coset $Nh$;

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

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

Viewed another way, suppose a finite group $G$, containing a non-trivial proper subgroup $H$, acts transitively on a finite set $\Omega$ with $\# \Omega \geq 2$, such that $H = \{ u \in G : \omega ^ { u } = \omega \}$ for some prescribed element $\omega \in \Omega$ and such that only the identity of $G$ leaves invariant more than one element of $\Omega$. Then $G$ is a Frobenius group with Frobenius complement $H$. Any element $n \neq 1$ of the Frobenius kernel $N$ acts fixed-point freely on $\Omega$, i.e. $\omega ^ { n} \neq \omega$ for each $\omega \in \Omega$.

There is a characterization of finite Frobenius groups in terms of group characters only. Namely, let $N$ be a subgroup of a finite group $G$ satisfying $\{ 1 \} < N < G$. Then the following assertions are equivalent:

a) statement 6) above;

b) $N$ is a normal subgroup of $G$ and $G$ is a Frobenius group with Frobenius kernel $N$. The step from b) to a) was known to Frobenius; the converse step with, in addition, $N$ normal in $G$ is surely due to Frobenius; however, the step from a) to b) with $N$ not necessarily normal in $G$ 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 $p$-modular Frobenius groups; see [a8]. Namely, let $K$ be a field of positive characteristic $p$. Then $G$ is a $p$-modular Frobenius group if it contains a non-trivial normal subgroup $N$ such that $K$ is a splitting field for the group algebra $K [ N ]$ and if one of the following (equivalent) statements holds:

A) every non-principal irreducible $K [ N ]$-module $V$ has the property that the induced $K [ G ]$-module $V ^ { G }$ is irreducible;

B) $C _ { G } ( x ) \leq N$ for every $p$-regular non-trivial element $x \in N$. Any $N$ featuring in A)–B) is a $p$-modular Frobenius kernel.

In 1959, J.G. Thompson [a9] showed that for a "classical" Frobenius group $G$, the Frobenius kernel $N$ 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 $p$-subgroup (cf. also Sylow subgroup) of a Frobenius complement $H$ of $G$ is cyclic (cf. also Cyclic group) when $p$ is odd, and cyclic or generalized quaternion if $p = 2$. He also proved that if $H$ 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 $p$-modular Frobenius groups. Namely, a $p$-modular Frobenius kernel $N$ is either solvable (cf. also Solvable group) or else $p = 2$ and any non–Abelian composition factor of $N$ is isomorphic to $\operatorname { PSL } ( 2,3 ^ { 2^t } )$ for some integer $t \geq 1$.

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

any Sylow $q$-subgroup of $G / N$ is cyclic whenever $q$ is relatively prime to $2 p$;

any Sylow $2$-subgroup of $G / N$ is cyclic or generalized quaternion if $p$ is odd. On the other hand, any non-trivial finite $t$-group ($t$ a prime number) is isomorphic to some quotient group $X / Y$, where $X$ is a suitable $t$-modular Frobenius group with $t$-modular Frobenius kernel $Y$. See also [a8].

Historically, finite Frobenius groups have played a major role in many areas of group theory, notably in the analysis of $2$-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 $N$ consist of the identity and those elements of the Frobenius group $G$ not occurring in any point stabilizer (cf. also Stabilizer). Contrary to the finite case, it is now not always true that $N$ is a subgroup of $G$. 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