4-Dimensional Elation Laguerre Planes Admitting Non-Solvable by Steinke G. F.

By Steinke G. F.

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We have G D AL D b 1 ; a2 S and the cyclic group hai=hzi acts faithfully on L. m n 1 D t 2 D s 2 D Œa; t  D 1; m 4; n 5; b t D (f) G D ha; b; t; s j a2 D b 2 m 1 n 2 i b s D b 1 ; a2q D b2 D z; b a D b 1C2 ; i D n m C 1 2; t s D m 2 i 1 a 2 2 t b; s D a b si. A/ L. A/. The order of G is 2 and G D ha; t; si. m; n/ (see Appendix 14). Here we have G D ha; b; t; s j m n 1 a2 D b 2 D t 2 D s 2 D Œa; t  D Œa; b D 1; m 2; n 3; b t D m 1 n 2 b s D b 1 ; a2 D b2 D z; t s D t b; as D a 1 z v ; v D 0; 1; and if v D 1; then m 3i.

G=L/ D 2 and so x induces an automorphism of order 2 on L such that t x D t u; ha2 ix D ha2 i and x centralizes the subgroup hvi of order 4 which is contained in ha2 i. a/). 2); in particular, in the case under consideration, m 5. a2 /x D a2 , we obtain ax D at i or ax D aut i . mod 2m 2 /; then aj D a or aj D au. a2 /x D a2 u, we get ax D avt i or ax D auvt i , where we set m 3 v D a2 . Indeed, then j Á 1 C 2m 2 so aj D av or aj D auv. a2 / 8 and so m 5. a/ D 2m 1 . at /t i . t / D ht i Q0 . Hence we may assume from the start that ax D at i , 32 Groups of prime power order 2 i D 0; 1.

T / D ht i Q, where Q is a generalized quaternion group of order 2m , m 3. We assume, in addition, that G has a normal four-subgroup U such that t … U . m; n/. m; n/, then we must have S D G. m; n/, then jG=S j Ä 2 and if, in addition, jG=S j D 2, then we have m D n. m; m/ and G acts transitively on the set of involutions in S U . Proof. Let E4 Š U G G with t 62 U . U / and then we have obviously t 62 T . This gives jG W T j D 2 and so G D ht iT . G/. G0 /. UG0 / W G0 j D 2, we have UG0 Ä G1 . G0 / D ht; zi that t has exactly two conjugates t and t z in G1 .

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