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MR0004042 (2,309c) 20.0X Brauer, R.; Nesbitt, C. On the modular characters of groups. Annals of Mathematics. Second Series 42 (1941), 556–590. The authors first summarize the theory of modular characters of a finite group G as previously developed by them [Univ. of Toronto Studies, Math. Series, no. 4, 1937]. If an element of G is defined to be p-regular when its order is prime to p, one of the important results in this theory is that the number of distinct absolutely irreducible modular representations of G is equal to the number of classes of conjugate p-regular elements of G. The principal problem considered in this paper is the reduction of an ordinary representation of G when the coefficients are taken in a field of characteristic p. It is proved that, if g = pag′, where (g′, p) = 1 and g is the order of G, then an ordinary irreducible representation Zi of degree zi ≡ 0 (mod p) remains irreducible as a modular representation and the character ζ of Zi vanishes for all elements of an order divisible by p. The theory of “blocks,” which is essential to the argument, cannot be explained here. As in the ordinary theory, the Kronecker product of two modular representations of G yields a new modular representation whose reduction is discussed. It is shown that the number of self-contragradient modular characters is equal to the number of self-reciprocal p-regular classes of G. Again, there is a modular theory corresponding to the ordinary theory relating the representations of G to those of a subgroup H. As would be expected, the analogy is very close. The paper concludes with a number of special cases and applications of the theory to GLH(2, p), SLH(2, p) and LF (2, p). G. de B. Robinson From MathSciNet, July 2016