Tame component of the Schur multipliers of finite groups of Lie type

Abstract

Let f be a finite field of characteristic p. Let .% be an almost f-simple, simply connected, algebraic group defined over f. For any f-subgroup A? of .Y, we shall denote by R(f) the finite group of f-rational points of Z, and shall consider Q/Z as an R(f)-module with trivial X(f)-action. Let G = F(f). Let J be the p-primary component of Q/Z, and J’ be the sum of all primeto-p primary components of Q/Z. Then Q/Z = JO J’, and the Schur multiplier H2(G, Q/Z) is the direct sum of H*(G, J) and H*(G,J’). The subgroup H2(G, J) is the p-primary component of the Schur multiplier of G; we shall call it the wild component of the Schur multiplier, and H2(G, J’), which is the sum of prime-to-p primary components of the Schur multiplier, the tame component. It is a classical result of Schur (41 that the tame component of the Schur multiplier of SL(2, 5) is trivial for any finite field 5. Griess has proved [3 1 that the Schur multiplier of SU(3, 5) is trivial. We shall use these two results and the Tits building of Y/f to give a new proof of the following theorem of Steinberg [6, 71 and Griess [ 31.