By A. A. Albert

ISBN-10: 082181401X

ISBN-13: 9780821814017

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O2t = 9; uniqueness still open. 10 Let F be the Galois field GF(q), q,+ 2, and let R be a two dimensional vector space over F, with basis elements 1, tr. (z) = z2 - az - b be an irreducible quadratic over F, and then define multiplication in R by: (i) x(u+ Iv) =xu+ tr'xv _1 (ii) if y1O,(x+ \y)(u+ rv)=)m- y ^vf(x) + I(yu-xv+av). Roughly, this corresponde to demanding that every element of R, except- ing elements in F, shall satisfy f(z) = O. The system R is always aproper left V-W system and the multiplication is associative only in the case that F = GF(3) andl(z) = 12 * 1.

R 8urino11o; aql a{€ru deur Jo uorleueldxa u€ sv 'palJsrles aq IIr^\ ^\olaq eA\ pa1;lcads s? r8qns lslsuoc'1 - u ueql ssal uolsuaulp leuolsuaulp-(I - u) aql 'I-P1 on1 fue (rrt) 'z-u' p roJ I tr ug ale8nfuoa ar? rSqns II" ]o las aql ]o sls]s observations lead to the following Definition conjugate 2. Let M be a group which contains a complete set of subgroups Pi (i = 1, . m) satisfying the postulation of Definition 1. Then the P, shall be called the points of a geometry f . The group M acts as a group of transformations on I' according to the rule that, for any element t of M, the map t(Pt) of Pt is defined by t(Pi) = tPit-1.

Albert, On non-associative division algebras, Trans. Amer. , vol. 72 (1952),296-309. 2. , Finite noncommutative division algebras, Proc. Amer. 3. , On the collineation groups associated with twisted fields, Math. , vol. 9 (1958), 928-932. Bull. CalEutfa-[Iath. Soc. (to appear). 4. Johannes Andr6, Uber nicht-Desarguessche Ebenen mit transitiver Translationsgruppe, Math. , vol. 60 (1954), 156-186. 5. ten in endlichen projektiven Ebenen, Arch. , voi 6 (1954), 29-32. 6. (1955), 15F16-0-. , Projektive Ebenen iiber Fastkijrpern, Math.

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