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Therefore (H, +, ·)is a near-domain and the proof is completed. 4)). 4), G is of type 0 if and only if char G = 2. 1)b J 2 * is a conjugate class, and hence all of its elements have the same order. For a= (1, 1), we have a E P, since a= (1, -1} (0,-1), and an= 2. 2) Let G be a sharply 2-transitive group, then we have: (i) G is of type 0 (ii) If G is of type 1, then {a} C J 2 , and ord a =char G, for every a E J 2 *. Further, if char G = p =I= 0, then each complement G0 contains a cyclic subgroup of order p - 1.

Now obviously G~ ~Goo. l)(a). § 11. KT·Fields and Sharply 3-transitive Groups Analogously to § 6 we show here that shaqjly 3-transitive groups can be completely characterized by means of transformations on a suitable algebraic structure. The structures introduced here are derived from the pseudofields defined by Tits in  and form the near-domains of Karzel, and hence the name KT-field. Definition: A KT-field is a quadruple (F, +,·,a), which satisfies the following axioms: KT 1: (F, +, ·) is a near-domain KT2: a is an involutory automorphism of the multiplicative group (F*, ·) which satisfies the jUnctional equation a(1 + a(x)) = 1- a(l + x), for all x E F\{0, -1}.

Db~p,1 ~ db,ada,b+1· Also for a, bEE we have: (II) Case 1. a= d1' b+a [(b +a)+ 1]. (II) db, a = 1 => 1 + (b + a) =(b + a) + 1 => b + (1 +a) = b + (a + 1) b + a E E. (I) Case 2. E. = 1 +(a+ b) (a+ b)+ d0 b =b + a => db ,a ;::: db+a,,1 =1 => b + a E Case 3. db, a ::f. 1 and d0 , b+l ::f. 1 =>db, a, d0 , b+l El: A, since AnD= 1. Thus § 9. a d~. b + 1 E A, since [F* : A] = 2, and this together with AnD= 1 implies db+a, 1 = 1, and hence b +a E E. 17). Now suppose IE I > 2 and c E E, c =I= 0, c =I= 1.