1 and the assertion holds for every smaller value of s. Since Aq * *6= 1, by Lemma 4.2, we get AqR~2= Sq2s-1(R1) + BR22. So n+1 Y 2i 2 2q QR2 = Ai (AqR~) 0 i2, it suffices to show Q2Q1 0 (mod I1). From [7, Theorem 2.2], we get X Q1 = xff11.x.f.fkk ff1+...+ffk=2k-2,ff X i=0 or power2of k-1 X 2 2 2 8 2k-1 2 = x1x2x43. .x.2k + x1x2x3x4. .x.k + R , sym sym P where denotes the sum of all symmetrized terms in x1, . .,.xk, and R is some sym polynomial, whose monomials are all of positive degree. By Lemma 2.5, R2 0 (mod I0). We obtain X k-1 k-1 Q1 (x1x2x43. .x.2k + x21x22x23x84. .x.2k ) (mod I0) sym X k-1 Sq2( x1x2x23x84. .x.2k ) (mod I0) sym X k-1 Sq2Sq1( x1x2x3x84. .x.2k ) (mod I0) sym Sq2Sq1(R1) (mod I0), P k-1 where R1 := x1x2x3x84. .x.2k . Writing Q1 = Sq2Sq1(R1) + Sq1(R2) for some sym R2 2 Pk, we get Q2Q1 = Q2Sq2Sq1(R1) + Q2Sq1(R2) R1Sq1Sq2(Q2) + R2Sq1(Q2) (mod I1) (by Lemma 2.4) R1Q0 (mod I1) (by Corollary 2.2). On the other hand, by [7, Theorem 2.2], we have X k-1 X k-1 Q0 = x1x22x43. .x.2k = Sq2( x1x22x23x84. .x.2k ) sym sym X k-1 = Sq2Sq2( x1x2x3x84. .x.2k ) = Sq2Sq2(R1). sym Therefore, Q2Q1 R1Q0 (mod I1) R1Sq2Sq2(R1) (mod I1) Sq2(R1)Sq2(R1) (mod I1) (by Lemma 2.4(b)) [Sq2(R1)]2 (mod I1) 0 (mod I1). Lemma B is proved. |___| References [1]L. E. Dickson, A fundamental system of invariants of the general modular li* *near group with a solution of the form problem, Trans. Amer. Math. Soc. 12 (1911), 75-98. CM* *P 95:18 [2]Nguy^e~n H. V. Hu'ng, The action of the Steenrod squares on the modular inv* *ariants of linear groups, Proc. Amer. Math. Soc. 113 (1991), 1097-1104. MR 92c:55018 5040 NGUY~^EN H. V. HU,NG AND TR^`AN NGO.C NAM [3]Nguy^e~n H. V. Hu'ng, Spherical classes and the algebraic transfer, Trans. * *Amer. Math. Soc. 349 (1997), 3893-3910. MR 98e:55020 [4]Nguy^e~n H. V. Hu'ng, The weak conjecture on spherical classes, Math. Zeit.* * 231 (1999), 727-743. MR 2000g:55019 [5]Nguy^e~n H. V. Hu'ng, Spherical classes and the lambda algebra, Trans. Amer* *. Math. Soc. 353 (2001), 4447-4460. [6]Nguy^e~n H. V. Hu'ng and F. P. Peterson, A-generators for the Dickson algeb* *ra, Trans. Amer. Math. Soc. 347 (1995), 4687-4728. MR 96c:55022 [7]Nguy^e~n H. V. Hu'ng and F. P. Peterson, Spherical classes and the Dickson * *algebra, Math. Proc. Camb. Phil. Soc. 124 (1998), 253-264. MR 99i:55021 [8]M. Kameko, Products of projective spaces as Steenrod modules, Thesis, Johns* * Hopkins Uni- versity 1990. [9]F. P. Peterson, Generators of H*(RP 1 ^ RP 1) as a module over the Steenrod* * algebra, Abstracts Amer. Math. Soc., No 833, April 1987. [10]S. Priddy, On characterizing summands in the classifying space of a group, * *I, Amer. Jour. Math. 112 (1990), 737-748. MR 91i:55020 [11]J. H. Silverman, Hit polynomials and the canonical antiautomorphism of the * *Steenrod algabra, Proc. Amer. Math. Soc. 123 (1995), 627-637. MR 95c:55023 [12]W. M. Singer, The transfer in homological algebra, Math. Zeit. 202 (1989), * *493-523. MR 90i:55035 [13]N. E. Steenrod and D. B. A. Epstein, Cohomology operations, Ann. of Math. S* *tudies, No. 50, Princeton Univ. Press, 1962. MR 26:3056 [14]R. M. W. Wood, Modular representations of GL(n, Fp) and homotopy theory, Le* *cture Notes in Math. 1172, Springer Verlag (1985), 188-203. MR 88a:55007 Department of Mathematics, Vietnam National University, Hanoi, 334 Nguy^en Tr* *~ai Street, Hanoi, Vietnam E-mail address: nhvhung@hotmail.com Department of Mathematics, Vietnam National University, Hanoi, 334 Nguy^en Tr* *~ai Street, Hanoi, Vietnam E-mail address: trngnam@hotmail.com