Analytic Geometry of Space by Virgil Snyder, Charles Herschel Sisam

By Virgil Snyder, Charles Herschel Sisam

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Then K = H or r(K} K. l). of K. Let j be a 2-element By making use of a well known theorem of Baer (see, for instance. H. LUneburg [61, pag. 20) and of lemma 1. we infer that j is a shear of axis V. OE(K} ~ 2r. e. E(K} order 2r _l. LEMr~A 3: Let p F 2. = K. f. automorphism group of Then oK PROOF: Assume that oK = pr or E(K} K = H. l) entails K = H. e. an even number: so N contains exactly one involution j. Moreover if . Thus ~ E K. -1 ~J = ~ • By lemma 1 j fixes (besides V = V n ~oo) U = U E ~oo.

3) In case e = 0 E = (resp. s3l -E . e'), from (sij) j = (sij) -1 lt 0) for each co 11 i nea t ion (s .. ) E 1: ( K) lJ -e' (resp. ) 1J By imposing K ~ Z(H), we infer so H = L(H). ) in K such that k3'l 42 lJ K. k Thus 1:(H) F L(H) and, on the ground of remarks 2 and 3, we deduce that only the case £ = -£' can hold. Furthermore, if (h .. ) E H - l:(H) we can set h.. = 0 for i < j and h21 = h43 F O.

In addi- On some translation planes admitting a Frobenius group 41 tion we claim that G fixes a component V of S and operates transitively on the .. p2r componen t s. If U is a subgroup of G, we will denote by L(U} and remalnlng E(U}, respectively, the group of collineations of U lying in the linear translation complement and the set of shears contained in U. The subgroup of col linea- tions of U which fix the set X of points (lines) of n is indicated by Ux ' First of all we will state the following LH1MA 1: H acts reguZarZy on S-{V} and N fixes a further component U of S.

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