The Coexistence of Elations and Homologies

27

Proof of Theorem 3. We shall construct a countable ascending chain of partial planes o^, 1^1, with the following properties: (1) ^^ скт^ and <т^ contains a rangle. If luJUU then o-j^cTi. (2) Each a^ is G-admissible, and for lujuU (^i is G-compatible with dj, (3) For each i, every point of (т. is joined to X, to Ä and to Д while every line of ст. meets x, a and b. (4) For each f, E acts on a. as a group of partial (Л, a) elations and H acts on ст^ as a group of partial {B, b) homologies. (5) If P, Q are distinct points not joined in cr._^, then P and ß are joined in d-. If /, m are distinct lines which do not intersect in (t._i , then / and m intersect in d..

We may take as a^ the partial plane n{^J which was constructed in Lemma 4 a), since it satisfies (2), (3), (4) and the first part of (1) by Lemma 4, and satisfies (5) and the second part of (1) vacuously. So suppose that, for some i^ 1, partial planes (7. which satisfy (l)-(5) have been constructed for 1 uj^i- Let ^ be the set of G- oi-bits of point-pairs which are unjoined in ст^, and let ^^ be the set of G-orbits of line-pairs which do not intersect in d.. We shall first construct cxf =5(т. such that af is G-admissible and G-compatible with (т-, such that af satisfies (3) and (4) and such that every pair of lines which do not intersect in a. intersect in af.

Well - order J^^ with order-type Я.. Denote the a'^ element of ^^ by L^, where l^a<A.. We shall construct partial planes a.^, for 0^a<A. with the following properties: (ly For each a, we have сг^^а^ „. If 0^а^)5<Я^, then a^^^^a^ß. (2У Each a, ^ is G-admissible. If 0^а^]в<'Я., then d.^ is G-compatible with fj. ^. (3)' For'each oc, every point of a-^ is joined to X, A and J5, while every line of a." intersects x, a and b. (4/ For each a, E acts on a.^ as a group of partial (Л, a) elations and Я acts on u.^ as a group of partial (J5, bj homologies. (5)' For each a, if {r^,s^}£L^, then r« and 5« intersect in d^,«.

We may take о.^ = а., since then (2)', (ЗУ and (4)' are satisfied because o. satisfies (2), (3) and (4), the first part of (1)' is satisfied trivially, and (5У and the second part of (ly are satisfied vacuously.

So suppose that for some ß with 0 < ^ < Я., partial planes a. ^ satisfying (1У-(5У have been constructed for all a with Oga<iS. Let d. „= IJ cr.^. Then a.^^ is

clearly a G-admissible partial plane which satisfies (3)' and (4)'. Moreover a^^ is G-compatible with tJ.^ for all a<iS, and for each a<)S and {r^, s^}£L^, the lines r^, s^ intersect in d. ^. Let {r^, s^}eL^. If r^ and s^ already intersect in a.^^, define d. ^ = d. p. If r^g and 5^ do not intersect in d.^, define <т^^ = 7г(^.^, {r^,s^}), structed as in Lemma 4b). In each case c.^ satisfies (iy-(5)'. This completes the induction step.

Now define (7*= \J a.^. Then a* satisfies (3) and (4), and every pair of

lines of cj. which do not intersect in <7. intersect in of. By well-ordering ^ and ing the dual procedure to af, we can now construct a^^^ satisfying (l)-(5). Finally,

let 71= Q (7f. By (1) and (5), тс is a projective plane. By (2) and (4), тс admits the

colhneation group G = E*H, where E acts as a group of {A, a) elations and Я as a group of (Б, b) homologies. This completes the proof of Theorem 3.

Corollary 1 There exists a projective plane which admits both an involutory elation

and an involutory homology.

Proof Take |E| = |Я| = 2 in Theorem 3.