164 A~CH. MATH. A Note on Finite Bol Quasi-Fields By ~'~ICHAEL J. K.~J-~HER I) 1. Introduction. A quasi-field Q is a Bol quasi-field ff a(b.ac) = (a. ba)c for all a, b, c e Q. An affine plane ~z is a Bol plane with respect to the non-parallel lines 1 and m if for every line k of d with k not parallel to 1 and m d has an in- volutory perspectivity with axis k which interchanges 1 and m. BUR⢠[2] gives the following connection between these two ideas: Theorem 1 (Bu~) . An a//ine plane d is a Bol plane with respect to the non-parallel lines I and m i /and only i /d is coordinatized by a Bol qua4i-/ield Q having 1 as its y-axis and m as its x-axis. A Bol quasi-field is proper if it is neither a nearfield nor a semi-field and a Bol plane is proper if its corresponding Bol quasi-field is proper. Bv~ [2] gives examples of infinite proper Bol quasi-fields. But no known finite quasi-field is a proper Bol quasi-field. The question of the existence of such an object is of interest since an affirmative answer would give new translation planes. In investigating this problem, KALL~ER and OSTRO~ [7] were able to prove that a finite proper Bol quasi-field, if it exists, has order p2r p a prime. In this note we shall show that p # 2. Specifically we prove : Theorem 2. A/ in i te Bol quasi-/ield Q o/ order 2 r, r a positive integer, is a near/ield. The proof makes use of a recent result due to HERI~G [5] giving the nature of a collineation group generated by elations fixing a point. We refer the reader to [4] for the basic ideas and terminology of the theory of projective planes. 2. Preliminary Results. We give here the machinery necessary for the proof of the theorem: Result I (GL~.~so~ [3]). Let p be a prime and G a finite permutation group on the set ~/[. I f for every point Q e J / there exists g e G such that g has order p and the only point fixed by g is Q, then G is transitive on all. 1) Partially supported by NSF Grant No. GP27101. Vol. XXIlI, 1972 Finite Bol Quasi-Fields 165 Result I I (H~r~G [5]). Let d be an affine translation plane of order 2 r, let S be the set of all elations fixing a point 0 e d , and let G be the collineation group gen- erated by S. One of the following statements hold: (i) G ~--- SL (2, 2s) for some integer s ~ 1. (ii) G ~--- Sz (2s), the Suzuki group of order 2 s, s odd and s ~ 3. (iii) G is elementary abelian. (iv) G contains a normal sub ,cup N having odd order and index 2. Result I I I (K~Jm~W~ [6]). Let d be a finite Bol plane of order 2 T coordinatized by a Bol quasi-field Q with respect to the lines I and m: I f d possesses a collineation a which neither fixes l and m nor interchanges them, then d is desargaesian and Q is a field. Returning to Theorem 1 for a moment, the lines of d can be represented as follows: 1 will be the line x --~ 0; i.e., every point on 1 has coordinates of the form (0, b), b e Q. The line m will be y = 0 ; i.e., every point on m has coordinates (a, 0), a e Q. Also it is well kno~m that every line k not parallel to l and m can be represented by an equation of the form y = mx q- b with m, b e Q, m * 0 ; this just means that a point on /c ~ill have coordinates (a, ma ~- b) for some a e Q. We will use this coordinatizing scheme in the proof of Theorem 2. 3. Proof of Theorem 2. First we give some general remarks. Let Q be a finite Bol quasi-field of order pr and let d be the associated translation plane coordinatized by Q in such a way that a point of 1 (Theorem 1) has coordinates (0, b) and a point of line m has coordinates (a, 0). Then for each a e Q, a . 0, d possesses the involutory perspectivity ~a: (x, y) --> (a- ly , ax) and also the collineation ra: (x, y) --> (ax, a - ly ) ([6]). Let G be the group generated by the ~a and let H be the group generated by the ~a. l~ote that H is a subgroup of G, since for every a* 0, Ta = ~0a~ where ~: (x, y) --> (y, x). In [7] it was shown that H a solvable ~oup implies Q is a nearfield except possibly - - ~2 34 in the cases pr 52, ~ , 112, 23 ~, (Theorem 3.2. The hypothesis given is that the autotopism gToup of ~ is solvable, but the proof only uses the fact that H is solvable). We turn now to proving the theorem. We may assume Q is not a field since every field is a nearfield. Hence z~ is not desarguesian, and Q has order ~eater than 2. Since Q has order 2r, by the preceding two paragraphs it is sufficient to prove the group G is solvable. Note that each ~a defined above is an elation with axis y ~ ax. Lemma 1. The coUineation group G is transitive on the lines l~ through (0, O) with l~ . l , b .m. Proof . Let J [ be the set of lines k through (0, 0) with k * l,/c * m. If/~ is the line y ----- ax, then the collineation ~ca defined above fixes/c and no other line of ~/[. Hence by Result I G is transitive on ..4,[. Lemma 2. G has a normal subgroup N o / index 2. 1 t}6 M. J . KALLAHER ARCH. MATH. P r o of. Let 2V be the subgroup consisting of the collineations in G fixing both 1 and m. Clearly 2Y is normal in G. ~u has index 2 since G = N w N~, where : (x, y) ~ (y, x). Lemma 3. G is the grou T generated by all elatio~s of d / ix ing (0, 0). P roo f . I t is sufficient to show that the elations ~Va are the only non-trivial elations fixing (0, 0). The line l can not be the axis of a non-trivial elation a; for if such a exists, then a would have to map m into a line k with k * l, k ~ m. By Result I I I d is desargxlesian -- a contradiction. Similarly m is not the axis of a elation. Consider a line k through (0, 0) with k =~ l, k =~ rn. We already know that ff k is given by the equation y = ax, then k is the axis of the elation ~a. Note that ~a interchanges l and m. I f a is another elation with axis k, then a has to map the line l into a line h. Since a is completely determined by what it does to l, h ~= m (for this would imply a -- 9a). I f h =~ l then Result I I I again ~ves rise to a contra- diction. Hence h ~ 1 and a is the identity collineation. Thus every line k, k * 1 and k * m, is the axis of only one non-trivial elation; this proves the lemma. We now proceed to prove Theorem 2. By Result I I and the last lemma we have four possibilities for G. G is not isomorphic to SL(2, 2s) since SL(2, 20 does not have a normal sub~oup of index 2 if s > 1 (ARTr~ [1], 13. 165). Similarly the Suzuki groups Sz(2 s) are simple (SuzuKI [8]) and hence G is not isomorphic to Sz(2s). By Lemma 1, I G] is divisible by 2 r -- 1 and it is clear that 2 divides ]G l . Hence G can not be elementary abehan. This leaves only (iv) of Result IV. Thus G is a group with a normal subgroup 2Y having odd order and index 2 in G. This implies G is solvable and the theorem is proved. Bibliography [1] E. ARTrg, Geometric Algebra. New York 1957. [2] R. P. BurN, Bol Quasi-fields and Pappos' Theorem. Math. Z. 105, 351--364 (1968). [3] A. M. GLEASON, Finite Fano Planes. Amer. J. Math. 78, 797--807 (1956). [4] M. H~L, Theory of Groups. New York 1959. [5] C. HE~I~G, On Elations of Translation Planes. To appear. [6] M. J. K.'.IZ-A~ER, A Note on Bol Projective Planes. Arch. Math. 20, 329--332 (1969). [7] M. J. KA~ER and T. G. OSTROM, Fixed Point Free Linear Groups, Rank Three Planes, and Bol Quasifields. J. Algebra 18, 159--178 (1971). [8] M. SuzuKI, On a Class of Doubly Transitive Groups. Ann. of Math. 75, 105--145 (1962). Auschrift des Autors: Michael J. Kallaher Department of Mathematics Washington State University Pullmann, Wash. 99163, USA Eingegangen am 31.3. 1971