On eigenvalues of the Cartan matrices for $p$-constrained groups

Let $p$ be a prime number. For $p$-constrained groups, we demonstrated that the Perron–Frobenius eigenvalue of the Cartan matrix of a $p$-block $B$ was bounded above by the order of its defect group. The proof employed block-theoretic methods, spectral bounds in modular representation theory, and structural $p^{\prime }$-reduction techniques. This conclusion extended a previously established inequality for $p$-solvable groups to the larger class of $p$-constrained groups. Finally, we presented computational results regarding the relationship between elementary divisors and eigenvalues of Cartan matrices for $p$-constrained groups that were not $p$-solvable.