1 - 16

What happens, if Xy is no longer primitive ? Then U/P acts reducibly on

P / Z ( P ) , by (2 iv) with P instead of Gj . Ь particular, the generator т that

has already appeared in the proof of (B4) acts as ( ^-i ) with С now being

contained in F rather than in F , . Thus C*^" = 1 • Ч may here be an even

p p'^

power of p or not. Again, G/P is abelian.

As a result, G/P can only be non-abelian when f^ is odd but f^^^^ is even.

Moreover , v, must be primitive (actually, even x„ = res x has to be so, as "■U о «jq

the argument given above implies).

Because of 2|f^/^ . UoPZ and thus Z(U)=Z(G). Hence Ü:=U/PZ is cyclic. But since G/P is supposed to be non-abelian, G := G/P Z is non-cyclic and for that reason already non-abelian. We infer that the 2-dimensional, cible, faithful F С G]-module V = P/Z(P) is irreducible. On the other

P hand, provided we suitably enlarge our base field F , the restriction of V to

thB cyclic group и of course splits into two 1-dimensional U-modules, and so each of them induces the original V . As V is faithful and as U is cyclic, they both are faithful as well.

Summing up we have shown that, over some extension field of Ж , V is

sr

induced by a faithful, 1-dinaensional U-module. As a matter of fact, G/U acts on и by the exponent -1 : remember that V is a symplectic module and that therefore the action of U is given by some matrix ( r"^) • Thus we have reached the following conclusion.

( 0 2) If. G/P is non-abelian, then f^ is odd, f / = 2 or 4, and, pondingly > ^^^rr/zr ^s stn even or odd divisor of p+1 , which is bigger than 2 .

'F / K

/ '^ e 4 -i -1

When f / = 2 , the group G/P Z is the group <т,а:т =l=a ,огта =т >

and has a generalized quaternion group as its Sylow 2-subgroup ; when f / =4 , it

- 1 -1 ^/-^

h the split eMension < т> x < a > with axa =r and e = ord т , 4 = ord a *

In any case, and independent of G/P being abelian or not, G/P Z is a sub-

2

group of SL(2, IF ) whose order divides p -1 but not p-] •