The Collineation Groups of Free Planes. II: A Presentation for the Group G 2

Projective planes generated by a quadrangle

“…Sandler, [7], [8], showed that F=(c~,/3, y) where (Q1Q2Q3Q4)cc=Q2Q1Q3Q4, (Q~Q2Q3Q,) /3 = Q2Q3'Q4Q1, (QxQ2Q3 Q,) y = Q1QsQ3QT, and that V has the presentation: <~, #, ~, io~ =/34= (~#p= ~= [(#~)~713 Mendelssohn,[4], considered the algebraic structure of the groups F, obtained by imposing on F the additional relation (cW)"= 1, n=2, ..., 6, and pointed out that for every odd prime p the little projective group L~ of the Desarguesian plane ~ of order p is a homomorphic image of F6, while L2 ~_F~. We study some geometrical questions related to this observation.…”

First-Order Model Theory of Free Projective Planes

“…[7,8,13,40,32]. In particular, there are numerous studies on the group of collineations and group of perspectivities of free projective planes, such as [2,21,33,34].…”

Collineations of freely generated planes

“…In [7] and [8] Sandier found the full collineation group of the free plane F2. It is an infinite group generated by three collineations and he found the relations between them.…”

“…If H= (KuA) then from Theorem 5.6. we can order A such that F (Kuh)=F(KuPt) for all i= 1 ..... n. Sandler has proved in[8] that if P~Im and him'where m and m' are elements of K then F(KuP~)=F~ (Kwh) if and only if we can express t, as t,=((((Ptrlnr2)ranr¢)...)rk63m' ) where rf is a point of F (K)ifi = 1(2)and a line of F (K) if i =0(2) for all i = 1, .... k. But since A is an orbit of we can write tj = ((((Pjr, 63 r2) r 3 63 r4)...) r k 63 m') for all j = 1, ..., n. Consequently if ~o is the perspectivity [m, rt] and cq is the perspectivity [r~, rt+l] i = 1, ..., k-1 and ~k is the perspectivity [rk, m'] then the projectivity ~=~o~1, ..., 0c k from m to m' is such that Ptc¢= h, i = 1, ..., n. Conversely, if such a projectivity exists then we can express all the tt in the form required by Sandler's result and hence F(Kutt)=F(KuPt) for all i = 1, ..., n and so II= (KuA) from Theorem 5.6. Now we shall study in more detail the orbits A of length n where ro(KwA)=2n.…”