By Janet E. Aisbett

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5) a0b + b0a= [:(x,y) •* 2a(x)b(y)]. 5 uses a representative of the class of the extension S :M >—> SL Z/4 -* GL Z/2. We will Jjust need the following: call Tn the subn n n group of strictly upper triangular matrices in GL Z/2, T Z/4 that in SL Z/4, and U that in M . If Jj : T -> GL Z/2 is the injection, the extension J n n n n :U >—> T Z/4-**T is easily seen to be j*(S). From this, we conclude that if J J K J n n n (|) is a canonical representative of S, im § \ (T x T ) <=• U . 6: Proposition, d ' x = 0.

AISBETT sets up the initial step from which the corresponding results for general G follow by induction on k. This occurs in the second section, where the module structure is also obtained. MOD p COHOMOLOGY OF ker(r : SL Z/p3 -*• SL Z/p) SI. All d** differentials and E** terms in this section refer to the LyndonSerre spectral sequence H*(G^; H*(Mn; z/p)) -> H*(G 3 ; z/p) 3 considered in the category of SL Z/p -groups and equivariant maps. 1: Notation. }. of M Z/p, {v.. , i ^ Jj, v.. = u.. }. of v(M Z/p) and ' n n 11 n* 13 i,j n -^ {v.

7 . we have H°(SL n Z/p; H 2 ^; X In particular, E ^ * = 0 2 # Z/p . 6). 3) there is an exact sequence •*\0 Oi£ 0*1 ^ ii im d ' H A M ->-> E^' . As A M has no SL Z/p-invariants, neither does . ,0,2 _, , ,0,2-,SL Z/p 0,2-,SL Z/p _, . Ap-, rT,0,2 im d ' . p in SL Z; similarly for the action of e.. ,A e Z/p. Therefore, [E0,2]SLnZ/p = [H 2 (N2 . 5: Proposition. H°(SLnZ/p2;H2(N2;Z/p)). If x e H5(SL Z/p2; Z/p) and i*x / 0 in H3(M ; Z/p), then R*x ^ 0 in H3(SL Z; Z/p). Proof. From the Z/p-spectral sequences associated to the commutative diagram: JANET E.

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