Character theory of $2$-Frobenius groups. - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T21:32:36Z http://mathoverflow.net/feeds/question/110134 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/110134/character-theory-of-2-frobenius-groups Character theory of $2$-Frobenius groups. Alexander Gruber 2012-10-20T04:47:07Z 2012-10-21T22:10:09Z <p>This is a crosspost of my (slightly longer) <a href="http://math.stackexchange.com/questions/215377/character-theory-of-2-frobenius-groups" rel="nofollow">question on MSE</a> since I'm not getting any responses there.</p> <hr> <blockquote> <p><strong>Definition.</strong> 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 <em>$2$-Frobenius</em>.</p> </blockquote> <p>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?</p> <p>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.</p> http://mathoverflow.net/questions/110134/character-theory-of-2-frobenius-groups/110138#110138 Answer by Geoff Robinson for Character theory of $2$-Frobenius groups. Geoff Robinson 2012-10-20T08:38:49Z 2012-10-20T08:38:49Z <p>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}?</p> <p>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.\$</p>