Let $V$ a vector space of dimension $n$ over the field $\mathbb{F}_p$ (with $p$ elements) and $\mathbb{F}$ the field with $p^n$ elements. Let $\mathcal{G}=Gal(\mathbb{F}|\mathbb{F}_p)$ and $\mathbb{F}^*=\mathbb{F}\setminus \{0\}$. Then we define $\Gamma(V)=\mathcal{G} \ltimes \mathbb{F}^*$, where the action of $\mathcal{G}$ on $\mathbb{F}^*$ is the natural one and $V$ is a $\Gamma(V)$-module by identifying the additive group of $\mathbb{F}$ with $V$.
In the study of the $p$-part of complex character degrees of a solvable group $G$ it is common to assume $\Phi(G)=1$, then $G$ splits over its Fitting subgroup $F=F_1(G)$, that is a completely reducible $G/F$-module. Call $\bar G=G/F$, then $V=Irr(F)$ is a completely reducible $\bar G$-module with $\bar G$ solvable. The general strategy is to reduce to the case where $V$ is irreducible and divide in two cases.
- $V$ is not quasi primitive, and then we can embed $\bar G$ in some particular wreath product and try here to bound from above the $p$-part of centralizers in $\bar G$ of elements of $V$.
- $V$ is quasi primitive, then in many situations we can embed $\bar G$ in $\Gamma (V)$, the semilinear group and study here the centralizer sizes.
Now, there exists a text, or a collection of sources, or at least something available on internet, where I can study semilinear groups in detail, in order to be able to work with them?
Meta: Should I post on math overflow?