Some results on the Cartan matrix of a Frobenius algebra

Let F be a field and A a Frobenius algebra over F. The Jacobson radical of A is denoted by J = J(A) and the kth socle of A by S-k(A). Let (A) over bar = A/J(k) or A/S-k(A). This article gives new interesting relations between the Cartan matrix of A and that of (A) over bar. From these results we prove that the Cartan matrix of A is diagonal if A/Soc(A) is a symmetric algebra. Let G be a finite group. If A is a block of F[G] with the above condition, then the Cartan matrix of A is (n), where n is the order of the defect group of A and the least integer such that J(n)(A) = 0.