Character theory of $2$-Frobenius groups.

2012-10-20T04:47:07Z

This is a crosspost of my (slightly longer) question on MSE since I'm not getting any responses there.

Definition. Let $G$ be a finite group and $F_1=\text{Fit}\,G$ and $F_2/F=\text{Fit}\left(G/F_1\right)$. If $F_2$ is a Frobenius group with kernel $F_1$ and $G/F_1$ is a Frobenius group with kernel $F_2/F_1$, we say that $G$ is $2$-Frobenius.

I have read about the characters of Frobenius groups in Isaacs and Huppert's books, but I have never seen $2$-Frobenius groups mentioned. Can anyone point me to some literature on the character theory of $2$-Frobenius groups?

Alternatively, does anyone know any theorems about Frobenius groups that could be adapted to $2$-Frobenius groups? I am especially interested in $2$-Frobenius groups where $F_1$ and $G/F_2$ are $p$-groups and $F_2/F_1$ is a $q$-group (for distinct primes $p$,$q$), but I would appreciate any representation theory at all which may help me better understand this class of groups.

Answer by Geoff Robinson for Character theory of $2$-Frobenius groups.

2012-10-20T08:38:49Z

I wonder if you have written exactly what you meant? Surely you mean that $F_{2}$ is a Frobenius group with kernel $F_{1}$ and $G/F_{1}$ is a Frobenius group group with kernel $F_{2}/F_{1}?

Assuming that is what you meant,there are lots of $2$-Frobenius groups, as you are probably aware. One family of examples, which is in a sense typical, is when you have a Frobenius group $H$ with Abelian Frobenius kernel $A$, and you take a faithful irreducible $FH$-module $V$ for $F$ a field of prime order. Then the semidirect product $VH$ is a $2$-Frobenius group according to your definition,because the irreducibility of $V$ ensures that $C_{V}(a) = 1$ for all non-identity $a \in A.$

Character theory of $2$-Frobenius groups. - MathOverflow

Character theory of $2$-Frobenius groups.

Answer by Geoff Robinson for Character theory of $2$-Frobenius groups.