The Cartan matrix of a certain class of finite solvable groups (Q1568357)

Let \(p\) and \(q\) be different primes, and let \(G\) be the group of affine semilinear transformations on the finite field with \(p^{pq}\) elements. The author computes the Cartan matrix of \(G\) in characteristic \(p\). He then uses this result in order to verify special cases of his conjecture that \(k(B)\), the number of irreducible characters in a \(p\)-block \(B\) of a \(p\)-solvable group, is bounded by \(\rho(B)\), the largest eigenvalue the Cartan matrix of \(B\).