Loewy structure for modules over semilinear groups (Q1906656)

Let \(G\) be the semilinear group \(\Sigma L(2,2^n)=\mathbb{Z}/n\mathbb{Z}\ltimes\text{SL}(2,2^n)\) (\(n\) odd), and \(k\) the algebraic closure of \(\mathbb{F}_2\). The author describes the irreducible and the projective indecomposable \(kG\)-modules, determines the Cartan integers as quotients of sums of algebraic integers, and the cohomology groups \(\text{Ext}^1_{kG}(S,T)\), for \(S\), \(T\) simple \(kG\)-modules. These results rely on heavy calculations using Clifford theory and the similar results of \textit{J. L. Alperin} [J. Pure Appl. Algebra 15, 219-234 (1979; Zbl 0405.20012)], concerning the group \(\text{SL}(2,2^n)\).