[For the entire collection see Zbl 0575.00010.]
The first Cartan invariant of a finite group G over an algebraically closed field K of characteristic \(p>0\) is the multiplicity of the trivial KG-module as a composition factor of its projective cover. This is the most complicated Cartan invariant to be computed. The author shows that in the case of \(p\geq 7\), the first Cartan invariants of the groups SL(3,p\({}^ n)\) and SU(3,p\({}^ n)\) are all equal to \(a^ n+b^ n+6^ n-2\cdot 8^ n\), where a, b are two roots of \(x^ 2-18x+48=0\). The methods involved are dependent on some results in the representation theory of the group SL(3,K), in particular, on the generic decomposition pattern of Weyl modules for SL(3,K). In his computation, the author constructs some digraphs and then he reduces the problem to compute the number of all closed walks of length n in these digraphs. Recently the author has also computed the Cartan invariant of \(Sp(4,p^ n)\) for large enough p.