By Bandini A.

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K[C ]-module i f Iker V is faithful, and from step 4 we know that Suppose that Icl. is faithful. From step 6 we know that that char k = p By is faithful. _-- = e/ker v. Ic I, ul G• is isomorphic with a direct summand of is algebraic. We conclude that V Step 8. V PE. is an VI G0 . is is a direct is algebraic, = PE. acts trivially on By step 7, V and - 47 - 1. C (E) therefore We now have shown that Z(G). El Thus (1)-(3) all P hold. 6)(d), vity of implies that every normal abelian subgroup of E structure of E the fonn primiti- P P is cyclic.

9), "K is a finite extension field of K. By the Wedderburn structure theorems for siluple algebras, where K is the center of A ces an antiautomorphism a oE ([CRI] SectLon 3n). A ~~ow we see that C\ indu- satisEying properties (i) and (iii). - 54 - Suppose x EG has image x in A and a E K, the center of A. Then (since a is an antiautomorphism) "- is an automorphism of (t so that all( = ~ + a I) = then + t I = o + I The proposition holds. = H0"'K A We now compute in the algebra we t E K Clearly, if ~ aj proving (il) .

C_ g. K[N]- isomorphism of \,v into - 60 - 4. Form InJllction and Clifford l s Theory We continue our examination of irreducible modules with a nonsingular form. 1) Hypothesis: group; and Assume that K is a finite field; G is a finite v is an irreducible V is a nonsingular classical bilinear form on V which is fixed by x V .. K G. In Section 1 part B(ii) we linked the induction structure of a symplectic module with the ten~or induction structure of a different irreducible mo- dule. The latter module was teusor primitive i f and only i f the symplectic module was fori.

### 3-Selmer groups for curves y^2 = x^3 + a by Bandini A.

by Mark

4.2