Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 41 (2000), No. 2, 557-568.

On the Classification of 16-dimensional Planes

Helmut Salzmann

Mathematisches Institut der Universität Tübingen Auf der Morgenstelle 10, D-72076 Tübingen e-mail: helmut.salzmann@uni-tuebingen.de

The concluding section Principles of classification of the book [18] on compact projective planes contains an outline of how a classification of 16-dimensional planes admitting a group Δ of dimension dim Δ ≥ h (where h is at least 35 ) might be accomplished. In the second part, proofs of several claims (of Proposition 87.4 in particular) have been indicated only very sketchily; some details will be supplied in the following, compare Theorem A below. Implications of Theorem A will be discussed elsewhere.

The automorphism group Σ of a compact plane P will always be taken with the compact-open topology. Only closed subgroups Δ of Σ will be considered, Δ is then a locally compact transformation group of the point space P.

Theorem L. If dim Δ ≥ 29, or if Δ is connected and dim Δ ≥ 27, then Δ is a Lie group.

This suffices for all classification purposes. A weaker result is given in [18] (87.1). For proofs see Salzmann [17] and Priwitzer-Salzmann [11].

From now on, assume that P is a compact 16-dimensional space, and that Δ is a connected Lie group. By the structure theory of Lie groups, there are 3 possibilities: Δ is semi-simple, or Δ contains a central torus subgroup, or Δ has a minimal normal vector subgroup Θ ≅ ℝ^{t}, compare [18] (94.26). In the first two cases, the results mentioned in [18] (87.2 and 3) have been improved in the meantime:

Theorem S. Let Δ be a semi-simple group of automorphisms of the 16-dimensional plane P. If dim Δ > 28, then P is the classical Moufang plane, or Δ ≅ Spin_{9}(ℝ, r) and r ≤ 1, or Δ ≅ SL_{3}ℍ and P is a Hughes plane as described in [18], §86.

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix}2000 Heldermann Verlag

The proof can be found in Priwitzer [9], [10].

Theorem T. Assume that Δ has a normal torus subgroup Θ ≅ 𝕋. If dim Δ > 30, then Θ fixes a Baer subplane, Δ′ ≅ SL_{3}ℍ, and P is a Hughes plane.

This is proved in Salzmann [15].

Hence only the case ℝ^{t} ≅ Θ ◁ Δ has to be considered. For convenience, the classical Moufang plane over the octonions will be excluded from the discussion. So-called stiffness theorems on the size of the stabilizer of a quadrangle play a decisive rôle: (‡) Suppose that the fixed elements of the connected closed subgroup Λ of Δ form a non-

degenerate subplane E.

(a) If dim Λ > 11, or if E is a Baer subplane, then Λ is compact.

(b) If Λ is compact, or if Λ is a Lie group and E is connected, then Λ ≅ G_{2} , or Λ ≅ SU_{3}ℂ, or dim Λ ≤ 7.

(c) If Λ is a compact Lie group and dim Λ < 8, then Λ ≅ SO_{4}ℝ or dim Λ ≤ 4.

(d) If Λ is a Lie group and E is a Baer subplane, then Λ is isomorphic to SU_{2}ℂ or dim Λ ≤ 1.

The first results are essentially due to Bödi [1], [2]. For (c) and (d) see Salzmann [13] and [18] (83.22).

Lemma 0. If ℝ^{t} ≅ Θ ◁ Δ and if dim Δ ≥ 24, then Δ fixes a point or a line, say a line W.

Proof. Grundhöfer-Salzmann [8], Proposition XI.10.19.

Theorem A. Assume that Δ is not semi-simple and that P is not a Hughes plane. If dim Δ ≥ 33, then, up to duality, Δ has a minimal normal subgroup Θ ≅ ℝ^{t} consisting of axial collineations with common axis W. Either Θ ≤ Δ_{[a,W]} is a group of homologies and t = 1, or Θ is contained in the group T = Δ_{[W,W]} of elations with axis W.

Remarks. This has been stated in [18], p. 587 under the stronger hypothesis dim Δ ≥ 36. The theorem does not assert that every given minimal normal subgroup is axial. The proof is fairly easy for t < 8 and rather involved for even t ≥ 8. The different cases will be treated in separate propositions. The result may well be true for even smaller dimensions of Δ, but a proof would become unreasonably complicated.

A group Ξ of collineations is called straight if each point orbit x^{Ξ} is contained in some line. The following result (Stroppel [19] Lemma 3 or Priwitzer-Salzmann [11], Th. B) is a clue to the proof of the existence of axial collineations:

Baer’s theorem. If Ξ is a straight subgroup of Δ, then Ξ is contained in a group Δ_{[z]} of central collineations with common center z, or the fixed elements of Ξ form a Baer subplane F_{Ξ} of P and Ξ is compact by (‡).

Corollary 1. If Π ≅ ℝ and Π is straight, then Π ≤ Δ_{[z,A]} for some center z and axis A.

Proof. Note that Π is not compact. By the dual of [18] (61.8), all elements in Π have the same axis. □

Corollary 2. If Θ ≅ ℝ^{t}, and if each one-parameter subgroup Π of Θ is straight, then Θ satisfies the assertions of Theorem A.

Proof. By [18] (61.7), the center map Θ \ {1l} → P is continuous, and the centers of all one-parameter subgroups of Θ form a compact and connected set Z. Commutativity of Θ implies that either Z is a single point, or Z is contained in the common axis W of all elements of Θ. The dual is also true. If there exist homologies in Θ, then t = 1 by [18] (61.2). □

Corollary 3. If Θ ≅ ℝ^{t} is a minimal normal subgroup of Δ and if some one-parameter subgroup Π of Θ is straight, then Θ satisfies the assertions of Theorem A.

Proof. Let Π ≤ Δ_{[z,A]} as in Corollary 1, and assume that Δ fixes the line W. Commutativity of Θ implies that z^{Δ} = z or z^{Δ} ⊆ A. If A = W, then Θ = Θ_{[A]} by minimality of Θ. If A ≠ W, then z^{Δ} ⊆ W and hence z^{Δ} = z. This case is dual to the first one. □

Lemma 1. Assume that the one-parameter subgroup Π of Θ is not straight. Then there is an orbit b^{Π} which generates a connected subplane; its closure will be denoted by E = ⟨b^{Π}⟩. For ϱ ∈ Π \ {1l}, the stabilizer Δ_{ϱ} in the action of Δ on Θ is the centralizer of Π, and the connected component Λ of Δ_{b,ϱ} induces the identity on E. Hence (‡) applies, and the dimension formula [18] (96.10) gives

(∗)  dim Δ = dim b^{Δ} + dim ϱ^{Δ_{b}} + dim Λ ≤ 16 + t + dim Λ.
□

Proposition 1. If t < 8 and if some one-parameter subgroup Π of Θ is not straight, then dim Δ ≤ 32.

Proof. Use Lemma 1. If Λ ≅ G_{2} , then dim E = 2 by [18] (83.24), and E consists of all fixed elements of Λ. Moreover, Λ acts trivially on Θ because Λ fixes ϱ and each non-trivial representation of G_{2} is at least 7-dimensional [18] (95.10). The connected component Θ_{b}^{1} of the stabilizer Θ_{b} is contained in the compact group Λ, hence Θ_{b}^{1} = 1l, dim Θ_{b} = 0, and dim b^{Θ} = t. Since Θ centralizes the group Λ, the fixed plane E of Λ is Θ-invariant. Consequently b^{Θ} ⊆ E, and it follows that t ≤ 2 and dim Δ ≤ 32. In all other cases dim Λ ≤ 8 by (‡), and dim Δ < 32. □

Proposition 1 and Corollary 2 imply that Theorem A is true in the case t < 8 and dim Δ > 32.

Lemma 2. Let ℝ^{t} ≅ Θ ◁ Δ and ℝ ≅ Π ≤ Θ. Assume that dim Δ > 32, and that the orbit b^{Π} is not contained in any line. Then Θ_{b} = 1l and 8 ≤ t ≤ 16. Moreover, the orbit b^{Θ} is not contained in any proper closed subplane of P, and Δ_{b} acts effectively on Θ.

Proof. Proposition 1 shows that t ≥ 8. Let F = ⟨b^{Θ}⟩ be the smallest closed subplane containing b^{Θ} . Note that F is connected and Δ_{b} -invariant, and that Θ_{b} fixes F pointwise. From (∗) it follows that 17 − t ≤ dim Λ. Obviously, b^{Θ} ⊆ F and Θ_{b}^{1} ≤ Λ. If dim F = 2, then t − 2 ≤ dim Λ and 15 ≤ 2 dim Λ. Therefore, dim Λ ≥ 8, and Λ is compact by (‡). Now Θ_{b}^{1} = 1l and dim b^{Θ} = t ≤ 2, a contradiction. If dim F = 4, then Δ_{b} induces on F a group Δ_{b}/Φ of dimension at most 8 (see [18] (72.8) and note that Δ_{b} ≤ Δ_{W} and b ∉ W). The kernel Φ of this action satisfies dim Φ > 8. The connected component of Φ is isomorphic to G_{2} by (‡), but this would imply dim F = 2, see [18] (83.24). Hence dim F ≥ 8 and F is a Baer subplane or F = P, for short, F ≤^{•} P. According to [18] (83.6), the group Θ_{b} is compact, and then Θ_{b} = 1l. Suppose, finally, that F < P. Then t = dim F = 8, and Δ_{b} induces on F a group Γ ≅ Δ_{b}/Φ of dimension dim Γ ≥ 14 (note that dim Φ ≤ 3 by (‡),(d). Hence dim ΓΘ ≥ 22, F is isomorphic to the quaternion plane, and Θ acts on F as a group of translations, see [18] (84.13) or Salzmann [14]. This contradicts the assumption that b^{Π} is not contained in a line. □

Note. For the proof of Theorem A in the case t ≥ 8 it is essential that Θ is chosen as a minimal normal subgroup. Write Ξ = Cs Θ for the centralizer of Θ. Then Γ = Δ/Ξ is an irreducible subgroup of GL_{t}ℝ. The structure of such groups is well known, compare [18] (95.6). In particular, the commutator subgroup Γ′ is semi-simple, and Γ is the product of Γ′ and the center Z of Γ. The irreducible representations of almost simple groups in dimension at most 16 are listed in [18] (95.10). Extensive use will be made of this list, compare also Bödi-Joswig [3]. Whenever X is an almost simple factor of Γ, the dimension of a minimal X-invariant subgroup of Θ divides t by Clifford’s Lemma [18] (95.5). If t is odd, then Γ′ is also irreducible and dim Z ≤ 1. Hence these cases are less difficult, they will be discussed next. A special argument is needed for large values of t.

Proposition 2. If t < 15 and t is odd, and if dim Δ > 32, then Θ satisfies the assertions of Theorem A.

Proof. As in Lemma 2, assume that the orbit b^{Π} is not contained in a line. Since Δ_{b} acts effectively on Θ, the Note shows that dim Γ′ ≥ 16, and Λ is mapped injectively into Γ. Now let t = 9. From (∗) it follows that dim Λ ≥ 8, and (‡) implies that Λ ≅ SU_{3}ℂ or Λ ≅ G_{2} . Therefore, dim Δ ≤ 39 and dim Γ ≤ 30. Moreover, Λ acts on a 6- or 7-dimensional subspace of Θ and fixes a complement, see the List [18] (95.10). By Clifford’s Lemma, Λ is properly contained in an almost simple factor Υ of Γ, and Υ acts effectively and irreducibly on Θ. Noting that 8 < dim Υ ≤ 30, the List shows that Υ ≅ PSL_{3}ℂ, but this group does not contain Λ. Hence t ∈ {11, 13}, and t is a prime number. Clifford’s Lemma implies that Γ′ is almost simple. Since dim Γ′ > 3, it follows from the List that dim Γ′ ≥ 55, an obvious contradiction. □

Lemma 3. A group Δ of dimension ≥ 31 fixes at most one point a ∉ W. Assume that t ≥ 12and that Θ|_{W} ≠ 1l. Let L be a line such that L ∩ W = u ≠ u^{Θ} . If L does not contain the exceptional point a, then Θ_{L} = 1l and dim L^{Θ} ≥ 12. Hence L^{Θ} is not contained in any proper closed subplane of P, and Δ_{L} acts effectively on Θ.

Proof. The first assertion follows immediately from (‡) and the dimension formula. Either ⟨L^{Θ}⟩ = Q is a closed subplane, or L^{Θ} is contained in a pencil 𝔏_{x}, and x^{Θ} = x. Suppose that x^{Δ} ≠ x. Then Θ_{u} induces the identity on the connected subplane D = ⟨x^{Δ}, u^{Θ}⟩. By the dimension formula, 12 ≤ t = dim u^{Θ} + dim Θ_{u}, and dim Θ_{u} ≥ 4. Since Θ_{u} is not compact, (‡) implies dim Θ_{u} ≤ 7, and dim u^{Θ} ≥ 5. Consequently, D = P and Θ_{u} = 1l. This contradiction shows that x^{Δ} = x is the unique point a. Hence L ≠ au leads to the first alternative, and Θ_{L} acts trivially on Q. Either dim Θ_{L} ≥ 8 or dim L^{Θ} > 4 and Q ≤^{•} P. In both cases, Θ_{L} is compact by (‡), and then Θ_{L} = 1l. Consequently, Q = P and Δ_{L} ∩ Cs Θ = 1l. □

The lemma leads to a useful modification of condition (∗):

Proposition 3. In the situation of Lemma 3, each one-parameter subgroup Π of Θ has some orbit b^{Π} which is not contained in any line. If u and L are chosen as above, let ϱ ∈ Π ≤ Θ_{u}, ϱ ≠ 1l. Then Θ_{u} ◁ H = Δ_{L}Θ, dim ϱ^{H} ≤ dim Θ_{u} ≤ 8, and

(∗∗) H < Δ, dim Δ/H ≤ 16 − t, and dim H ≤ 16 + dim ϱ^{H_{b}} + dim K ≤ 24 + dim Λ,

where K denotes the connected component of H_{b,ϱ} = Δ_{b} ∩ Cs_{H}Π and Λ has the same meaning as in Lemma 1.

Proof. Because Δ_{L} fixes the point u and Θ is commutative, Θ_{u} is invariant in H = Δ_{L}Θ. By assumption, Θ_{u} < Θ and Θ is a minimal normal subgroup of Δ. Moreover, dim Θ_{u} ≥ t − 8 ≥ 4. Consequently, H ≠ Δ. From Δ_{L} ∩ Θ = 1l it follows that dim H = dim Δ_{L} + t, and the dimension formula shows dim Δ = dim Δ_{L} + dim L^{Δ} ≤ dim Δ_{L} + 16. Therefore, dim Δ − dim H ≤ 16 − t. Since dim b^{H} ≤ 16, the last inequality in (∗∗) is immediate from the dimension formula. □

Proposition 4. Suppose that dim Δ ≥ 31 and that u^{Θ} ≠ u ∈ W = W^{Δ} . If dim Λ > 8, then Λ ≅ G_{2} by (‡), and the centralizer X = Cs_{Θ}Λ has dimension at most 2. Moreover, t ∈ {1, 2, 8, 9}.

Proof. Use the same notation as in Proposition 3. By assumption, Π ≤ X and dim X ≥ 1. The orbit b^{X} is contained in the 2-dimensional subplane E of the fixed elements of Λ. Obviously, X_{b}^{1} ≤ Θ ∩ Λ = 1l. Therefore, dim X ≤ 2. From [18] (95.3 and 10) it follows that the complement of X in Θ has a dimension divisible by 7. Hence the proposition is true unless t ≥ 15. Then Proposition 3 applies, and the first statements of (∗∗) exclude the possibility t = 16. In the case t = 15, condition (∗∗) implies that dim Δ ≤ 39. Hence Δ induces on Θ an irreducible group Γ of dimension at most 24. According to the note, Γ′ is irreducible on Θ, and Λ is properly contained in Γ′. By Clifford’s Lemma, Γ′ is almost simple. Inspection of the List [18] (95.10) shows that there is no group with these properties. □

Propositions 3 and 4 imply:

Corollary 4. If t ≥ 12, then dim Δ ≤ 48 − t or Θ satisfies the assertions of Theorem A.

Corollary 5. If t = 16 and dim Δ > 32, then Θ = T is a transitive elation group.

Proposition 5. If t = 15 and dim Δ > 32, then Θ is a group of elations.

Proof. Assume that u^{Θ} ≠ u for some point u on the fixed line W of Δ. Then dim Δ = 33 by Corollary 4, and Propositions 3 and 4 give dim H = 32 and Λ ≅ SU_{3}ℂ. Moreover, (∗∗) implies dim ϱ^{H} = 8 for each admissible ϱ. Hence Δ_{L} is transitive on Θ_{u} ≅ ℝ^{8} . With [18] (96.22) or the List (95.10) it follows that the commutator subgroup Υ of Δ_{L} is isomorphic to SU_{4}ℂ. According to the Note and to Clifford’s Lemma in particular, Υ is properly contained in an almost simple factor of Γ ≅ Δ/Cs Θ. This implies dim Γ ≥ 21 and dim Δ ≥ 36, a contradiction. □

Lemma 4. Let p and q be prime numbers. A semi-simple irreducible subgroup G of SL_{pq}ℝ has not more than two almost simple factors.

Proof. G is either almost simple or a product of two proper semi-simple factors A and B such that B ≤ Cs A. Let U be a minimal A-invariant subspace of V = ℝ^{pq}. If U = V , then B ≤ ℍ^{×} by Schur’s Lemma [18] (95.4). Hence 4 | pq = 4 and G ≅ SO_{4}ℝ. If U < V , however, then A acts effectively on U (because the fixed elements of the kernel of the action of A on U form a G-invariant subspace of V ). Clifford’s Lemma shows that dim U ∈ {p, q} and that A is almost simple.

Alternative proof (suggested by the referee). All semi-simple irreducible subgroups G of SL_{t}ℝ have been determined by Dynkin [5/6] Th. 1.5: either G is maximal in SL_{t}ℝ or in a symplectic or orthogonal group, and the claim follows from Theorems 1.3 and 1.4, or G belongs to a long list of exceptions, but these are even almost simple. □

The following well-known theorem will be needed several times:

Complete reducibility. If a semi-simple group G acts on a real vector space V , then each G-invariant subspace of V has a G-invariant complement in V .

This follows directly from an analogous result about representations of semi-simple Lie algebras, see e.g. Bourbaki [4] I.6.2 Theorem 2, p. 52 or Freudenthal-de Vries [7] § 35 or § 50.

Proposition 6. If t = 14 and dim Δ > 32, then Θ is a group of elations.

Proof. Assume that Θ does not act trivially on W. Put again Ξ = Cs Θ.

(a) Corollary 4 shows that dim Δ ≤ 34. According to Lemma 2 and the Note, the stabilizer Δ_{b} acts effectively on Θ and may be considered as a subgroup of Γ = Δ/Ξ, and the semisimple commutator group Γ′ satisfies 15 ≤ dim Γ′ ≤ 20. No almost simple group with dimension between 15 and 20 has an irreducible representation in dimension 7 or 14.

Therefore, Lemma 4 implies that Γ′ is a product of exactly two almost simple subgroups A and B. Let dim A ≤ dim B. Then dim B ≥ 8, and B acts effectively and irreducibly on ℝ^{7} . By the List, dim B = 14, and B is one of the two simple groups of type G_{2} . (Later it will be seen that B is in fact the compact form.) As in the proof of Lemma 4, it follows that A acts effectively on ℝ^{2} . Consequently, A ≅ SL_{2}ℝ and dim Γ′ = 17, moreover, the center Z of Γ consists of real dilatations of Θ, see [18] (95.6).

(b) The last statement gives dim Γ ≤ 18 and dim Ξ ≥ 15. On the other hand, dim Ξ ≤ 16 since dim Δ/Ξ ≥ dim Δ_{b} ≥ dim Δ − 16. Therefore, Ξ is contained in the radical \sqrt{Δ}, and a maximal semi-simple subgroup Ψ of Δ is locally isomorphic to Γ′.

(c) The group Δ has a subgroup Δ̂ of codimension 3 which induces on Θ the group BZ and acts on a 7-dimensional subgroup N of Θ. Note that dim Δ̂_{b} ≥ 14. If Δ̂_{b} is transitive on N, then it is also transitive on the 6-sphere consisting of the rays in N, and Δ̂_{b} contains the compact group G_{2} , compare [18] (96.19 and 22). If Δ̂_{b} is not transitive on N, then for some ϱ ∈ N the connected component Λ̂ of Δ̂_{b,ϱ} is at least 8-dimensional, and (‡) shows that Λ̂ ≅ SU_{3}ℂ. This group is not contained in the non-compact group G_{2}(2). Therefore B is compact, and steps (a) and (b) imply Γ′ = A × B ≅ SL_{2}ℝ × G_{2} ≅ Ψ.

(d) Because Δ_{b} ↪ Γ and dim Γ/Δ_{b} ≤ 1, it follows that G_{2} ≅ B ↪ Δ_{b}. One can now conclude that Δ_{b} acts irreducibly on Θ. In fact, the action of B and complete reducibility force a proper Δ_{b} -invariant subgroup N of Θ to be 7-dimensional. Lemma 1 with ϱ ∈ N gives dim Δ_{b,ϱ} ≥ 10 and then Λ ≅ G_{2} , but this contradicts the fact ϱ^{Λ} = ϱ. As a consequence of [18] (95.6(b)), even the action of the semi-simple group Δ_{b}′ on Θ is irreducible, hence dim Δ_{b}′ = 17 and Δ_{b}′ ≅ Ψ.

(e) In particular, the involution in A corresponds to an involution α in the center of Δ_{b}. By [18] (84.9), the group B ≅ G_{2} cannot act on a Baer subplane. Consequently, α is a reflection, see [18] (55.29). Because of (‡), the group A acts effectively on the 2-dimensional plane E = F_{B} of the fixed elements of B. If A would fix a flag in E, then A would be solvable by [18] (33.8). Therefore, b and W are the only fixed elements of A in E, and α ∈ Δ_{[b,W]} . Since Θ_{b} = 1l, it follows from [18] (61.19b) that α^{Θ}α is a 14-dimensional subset of T = Δ_{[W,W]} . Note that ΘT ≤ \sqrt{Δ} and that dim \sqrt{Δ} ≤ 17. Minimality of Θ implies Θ ≤ T. □

Proposition 7. If t = 10 and dim Δ > 32, then Θ is a group of elations.

Proof. From (∗) and Proposition 4 one obtains 17 ≤ dim Δ_{b} ≤ dim ϱ^{Δ_{b}} + dim Λ ≤ 10 + 8. Either Δ_{b} is transitive on Θ, or Λ ≅ SU_{3}ℂ and dim ϱ^{Δ_{b}} = 9 for some ϱ ∈ Θ. In the first case, Δ_{b} would contain the group SU_{5}ℂ, and dim Δ_{b} would be to large, compare [18] (96.16–22). Similarly, Δ_{b} cannot be transitive on a 9-dimensional subspace of Θ. Hence Δ_{b} acts effectively and irreducibly on Θ. Since each non-trivial representation of SU_{3}ℂ on Θ is either 6- or 8-dimensional, it follows from Clifford’s Lemma that Λ is not normal in Δ_{b}. Therefore, Λ is properly contained in an almost simple and irreducible factor X of Δ_{b}′, and X = Δ_{b}′ by Schur’s Lemma [18] (95.4). The List shows that Δ_{b}′ must be locally isomorphic to SL_{4}ℝ or to SL_{2}ℍ, but these groups do not contain SU_{3}ℂ. □

The only remaining cases t = 8 and t = 12 are more difficult. If the given group Θ does not consist of elations, it will be shown that some other normal vector group Θ̃ of Δ satisfies the conditions of Theorem A.

Proposition 8. If t = 8 and dim Δ > 32, then either Θ or some minimal normal subgroup Θ̃ ≅ ℝ^{7} consists of elations.

Proof. (a) According to Corollary 3, we may assume that for each one-parameter subgroup Π < Θ there is some point b such that b^{Π} generates a subplane. Lemma 1 then shows that the connected component Λ of Δ_{b} ∩ Cs Π has dimension at least 9, and Λ ≅ G_{2} by (‡). Consider the action of the connected component B of Δ_{b} on the 7-sphere S consisting of the rays in Θ, and let r, r′ denote the two opposite rays contained in Π. Since dim B ≥ 17 and dim B_{r}/Λ ≤ 1, it follows that r^{B} is a connected set of positive dimension. For each s ∈ S \ {r, r′}, the orbit s^{Λ} is a 6-sphere, and r^{B} is a connected union of Λ-orbits. Consequently, r^{B} contains an open neighbourhood of r in S, and r^{B} is open in S, see also [18] (96.25). The dimension formula implies dim B/Λ ≥ 7 and 21 ≤ dim Δ_{b} ≤ 22.

(b) In particular, r^{Δ} is open in S, and this is true for each ray r ∈ S because step (a) is valid for an arbitrary choice of r. Therefore, Δ acts transitively on S. Put Ξ = Cs Θ as in the Note. Then Γ = Δ/Ξ is the effective group induced by Δ on Θ. Remember from Lemma 2 that Δ_{b} is embedded into Γ. Step (a) and Lemma 1 imply 21 ≤ dim Γ ≤ 30. According to [18] (96.19), a maximal compact subgroup Φ of Γ acts transitively on S, and from [18] (96.20) it follows that Φ is isomorphic to a subgroup of SO_{8}ℝ. Moreover, Φ has a subgroup Λ ≅ G_{2} . There are only two groups Φ which satisfy these conditions, viz. Φ ≅ Spin_{7}ℝ and Φ ≅ SO_{8}ℝ, see [18] (96.21 and 22). The centralizer of Φ in GL_{8}ℝ is isomorphic to ℝ^{×}, and the remarks in the Note show that Γ is the product of Φ = Γ′ and the center Z of Γ. By [18] (94.27), the group Δ contains a subgroup Ψ which is a covering group of Γ′. In fact, Ψ is simply connected, since Δ cannot contain a group SO_{7}ℝ, see [18] (55.34 or 40).

(c) Assume that dim Γ′ > 21. Then Ψ ≅ Spin_{8}ℝ, and the center of Ψ contains 3 reflections α, σ, and ασ with centers a, u, and v. By (‡), the stabilizer ∇ of the triangle {a, u, v} satisfies dim ∇ ≤ 30, and Ψ ≤ ∇ < Δ. It suffices to consider the case Δ = ΨΘ. Note that Δ is not transitive on W (otherwise Δ would induce on W the group SO_{9}ℝ). On W \ v the action of Ψ is equivalent to a linear action, and for each z ≠ u, v the orbit z^{Ψ} is a 7-sphere, compare [18] (96.36). Hence z^{Δ} is open in W whenever z^{Δ} ≠ z^{Ψ} , see also [18] (96.25). If v^{Δ} = v, then u^{Δ} = u^{Θ} is open in W, and σ^{Θ}σ = {ϑ^{−1}ϑ^{σ} | ϑ ∈ Θ} would generate a transitive group of elations with axis av in Θ. Therefore u^{Θ} ≠ u and v^{Θ} ≠ v, and W contains some orbit z^{Δ} = Z ≈ 𝕊_{7} . This leads to a contradiction as follows: Since z^{Θ} is Ψ_{z} -invariant, the argument of [18] (96.25) shows that either z^{Θ} = z or z^{Θ} is open in Z. Because Ψ is transitive on Z and Θ is normal, all Θ-orbits in Z are equivalent. The commutative group Θ cannot be transitive on Z. Therefore the orbits are points, Θ|_{Z} = 1l, and Θ would act freely and transitively on az \ {a, z} ≈ ℝ^{8} \ {0} which is impossible.

(d) The last steps imply Ψ ≅ Γ′ ≅ Spin_{7}ℝ. Since 21 ≤ dim Δ_{b} ≤ dim Γ ≤ 22, and since Spin_{7}ℝ has no subgroup of codimennsion 1, the covering group Ψ of Γ′ can be chosen in Δ_{b}. Hence Δ_{b}′ ≅ Spin_{7}ℝ. Now it is not difficult to determine the structure of Δ. Put again Ξ = Cs Θ. Lemma 2 implies Ξ_{b} = 1l and dim Ξ ≤ 16. On the other hand, the bound on dim Γ gives dim Ξ ≥ 11. Consider the group Υ = Ξ ∩ Cs Ψ and note that Λ ≤ Ψ. The orbit b^{Υ} is contained in the 2-dimensional fixed plane F_{Λ} . Consequently, dim Υ ≤ 2. Let Ψ act on the Lie algebra 𝔩 Ξ. By complete reducibility, 𝔩Θ has an invariant complement 𝔫 in 𝔩 Ξ, and dim 𝔫 > 2. If dim 𝔫 < 7, the representation of Spin_{7}ℝ on 𝔫 is trivial and 𝔫 = 𝔩Υ, a contradiction. Hence dim 𝔫 ≥ 7 and dim Ξ ∈ {15, 16}. Because Ψ∩Ξ ≤ Ξ_{b} = 1l, we may assume that Δ = ΨΞ.

(e) The central involution σ of Ψ is a reflection, its axis K is different from W (or else σ^{Θ}σ = Θ would consist of elations). If dim Ξ = 15, then 𝔫 ≅ ℝ^{7} and Ψ induces on 𝔫 the group SO_{7}ℝ. Therefore 𝔫 is the Lie algebra of N = Ξ ∩ Cs σ, moreover, Ξ is a vector group and N is Δ-invariant. Proposition 1 shows that Θ̃ = N is an elation group as required.

(f) Finally, let dim Ξ = 16. Because Ξ_{b} = 1l, the orbit b^{Ξ} is open in P by [18] (96.11(a)). Note that b^{σ} = b. If ξ ∈ Ξ and b^{ξ} ∈ K, then ξσξ^{−1}σ ∈ Ξ_{b} = 1l and σξ = ξσ. This gives Ξ_{K} = Ξ ∩ Cs σ and dim Ξ_{K} = 8, moreover, Ξ = Ξ_{K} × Θ. Under the action of Ψ, the group Ξ_{K} splits into a one-parameter group and a 7-dimensional vector group Θ̃ on which Ψ induces a group SO_{7}ℝ. Obviously, Θ̃ is ΨΞ- and hence Δ-invariant, and Θ̃ is an elation group, again by Proposition 1 and Corollary 2. □

Proposition 9. If t = 12 and dim Δ > 32, then Δ has a normal vector subgroup Θ̃ consisting of elations. (Θ̃ may be different from Θ).

Proof. Assume that Θ is not contained in the elation group T = Δ_{[W,W]} , and use the notation introduced in Proposition 3.

(a) Propositions 3 and 4 imply that 29 ≤ dim H ≤ 32. Consider a minimal H-invariant subgroup M of Θ_{u} and let 1l ≠ ϱ ∈ Π ≤ M. Then H and Δ_{L} act irreducibly on M ⊃ ϱ^{H} , and dim ϱ^{H} ≥ 5 by (∗∗). Remember from Lemma 2 that Θ_{b} = 1l. Consequently, dim b^{M} > 4 and ⟨b^{M}⟩ = B is at least 8-dimensional (B ≤^{•} P). Statement (d) of (‡) shows that dim(H_{b} ∩ Cs M) ≤ 3, and H induces on M a group of dimension at least 10. If K denotes again the connected component of H_{b} ∩ Cs Π, then dim K ≤ 8 by Proposition 4. The following will be shown in the next steps: M = Θ_{u} ≅ ℝ^{8} and Δ_{L} does not act effectively on M.

(b) If M ≅ ℝ^{5} , then (∗∗) and (‡) imply K ≅ SU_{3}ℂ. This group does not admit a nontrivial representation in dimension < 6. Hence K ≤ Cs M contrary to what has been stated in (a).

(c) In the case M ≅ ℝ^{6} , the same argument shows that dim K < 8 (note that the action of K on M is not irreducible, since Π is K-invariant). Consequently, dim H_{b} = 13 by (∗∗), moreover, dim ϱ^{H_{b}} = 6 for each choice of ϱ, and H_{b} acts transitively on M \ {1l}, see [18] (96.11(a)). Let Ψ = H_{b}|_{M} denote the effective group induced on M. Since Ψ is irreducible and dim Ψ ≤ 13, the List of representations shows Ψ′ ≅ SU_{3}ℂ, and dim Ψ = 10. Hence H_{b} has a normal subgroup Φ ≅ SU_{2}ℂ acting trivially on M, cf. step (a) and (‡)(d). The involution ω ∈ Φ fixes a Baer subplane F_{ω} and the orbit b^{M} is contained in the pointset F of F_{ω} = B. Since H_{b} ≤ Cs ω, the group Ψ′ acts non-trivially on B and, therefore, on S = F ∩ W ≈ 𝕊_{4} , but this contradits Richardson’s theorem [18] (96.34).

(d) Finally, let M ≅ ℝ^{7} . Steps (b) and (c) imply that H_{b} acts irreducibly on M, and (∗∗) shows that dim H_{b} ≤ 15. Again, Ψ = H_{b}|_{M} is at least 10-dimensional. According to the Note, Ψ′ is an almost simple group of type G_{2} . By complete reducibility, M has a Ψ′ -invariant complement N ≅ ℝ^{5} in Θ. If ⟨b^{M}⟩ = B is a Baer subplane, then Ψ′M is a 21-dimensional automorphism group of B, and B is isomorphic to the quaternion plane H, see Salzmann [12], cp. also [18] (84.21(b)) or Salzmann [14], but H does not admit a group of type G_{2} . Hence ⟨b^{M}⟩ = P and Ψ′ is a subgroup of H_{b}. The List shows that Ψ′ centralizes N. Therefore, Ψ′ induces the identity on b^{N} . Because ⟨b^{M}⟩ = P, there is some point c ∈ b^{M} such that ⟨b^{N}, c⟩ = P. Consequently, Ψ′_{c} = 1l, but dim Ψ′_{c} ≥ 7, a contradiction proving the first statement at the end of step (a).

(e) Suppose now that Δ_{L} acts effectively on M ≅ ℝ^{8} . By assumption, this action is irreducible. The structure theorem [18] (95.6(b)) shows that the commutator subgroup Υ = Δ_{L}′ is semi-simple and that the center Z of Δ_{L} consists of real or complex dilatations of M. There are two possibilities: (i) the action of Υ on M is irreducible, and (ii) M is a direct sum of two 4-dimensional subspaces M_{ν} and Υ acts equivalently and effectively on the spaces M_{ν}. Note that dim Δ_{L} ≥ 17 and dim Υ ≥ 15. The action of the semisimple group Υ on Θ is completely reducible. Hence there is a Υ-invariant complement N ≅ ℝ^{4} of M in Θ. In case (i), the group N is even Δ_{L} -invariant: in fact, for each ζ ∈ Z the subspace N^{ζ}N is invariant under Υ and at most 8-dimensional, hence it has trivial intersection with M. Choose a one-parameter group E in N and a point p such that ⟨p^{E}⟩ = E is a subplane. Remember that H = Δ_{L}Θ has dimension at least 29. Consider the connected component K̃ of H_{p} ∩ Cs E. The dimension formula gives dim K̃ ≥ 9, and (‡) shows that K̃ ≅ G_{2} , but then Proposition 4 would imply t ≤ 9. Case (ii) also leads to a contradiction: Υ is semi-simple and effective on ℝ^{4} , therefore, Υ ≅ SL_{4}ℝ and dim Υ = 15, moreover, Z ≅ ℂ^{×} and Z has a subgroup P ≅ ℝ consisting of real dilatations. The group Ĥ = ΥPΘ acts on M_{ν} ≅ ℝ^{4} . Choose ϱ ∈ M_{ν} and write K̂ for the connected component of Ĥ_{b,ϱ}. Then dim K̂ ≥ 8, and K̂ ≅ SU_{3}ℂ by (‡), but the latter group is not contained in the maximal semi-simple subgroup Υ of Ĥ. This proves the last claim of step (a).

(f) The group M ≅ ℝ^{8} acts transitively on the affine line pencil L_{u} \ {W } = L_{u}^{∗} ≈ ℝ^{8} : in Lemma 3 it has been shown that Θ_{L} = 1l for each line L ∈ L_{u}^{∗} with at most one exception au. By [18] (96.11), each non-trivial orbit L^{M} is open in L_{u}^{∗} and homeomorphic to ℝ^{8} . Hence M is sharply transitive on L_{u}^{∗} or on L_{u}^{∗} \ {au}, but the latter space is not contractible.

(g) According to (e) there is an element α ≠ 1l in Δ_{L} ∩ Cs M, and (f) implies that α fixes each line in L_{u}. Consequently, α is an axial collineation with center u and some axis A. If α is an elation, i.e. if u ∈ A, then A = W (since α^{M} = α), and α^{Θ} is a 4-dimensional subset of T. Because the elements in α^{Θ} have different centers, T is commutative. Any minimal invariant subgroup of T may be chosen as the group Θ̃.

(h) Now let α ∈ Δ_{[u,A]} be a homology. Since u^{Δ} ≠ u, Hähl’s results on the generation of elations by homologies can be applied. In its simplest form, Hähl’s theorem says the following: (•) If Γ is a Lie subgroup of Σ_{A}, if Γ_{[c,A]} ≠ 1l for some center c ∉ A, and if E is the group of elations in Γ with axis A, then dim E = dim c^{Γ} , see [18] (61.20) for a proof. Note that commutativity of E is not known if c^{Γ} is contained in a line.

(i) Suppose that Θ ≤ Γ = Δ_{A}, and put A ∩ W = v. Then (•) shows that dim Δ_{[v,A]} ≥ 4. Moreover, dim u^{Θ} = 4 implies that the commutator set [α, Θ] = {α^{−1}α^{ϑ} | ϑ ∈ Θ} ⊆ Θ_{[A]} is at least 4-dimensional. Since Θ is commutative, all elements in Θ_{[A]} have the same center z ∈ W, and dim Θ_{[A]} ≤ 8. By [18] (61.2), a group of homologies has a compact subgroup of codimension at most 1, but Θ is a vector group. Hence Θ_{[A]} = Θ_{[v,A]} consists of elations. Because Θ is a minimal normal subgroup of Δ, the group Θ_{[A]} is not normal, and, therefore, A^{Δ} ≠ A. Since A^{Θ} = A and Θ is normal, Θ fixes each line in the orbit A^{Δ} . On the other hand, all fixed lines of the elation group Θ_{[A]} pass through the center v. Consequently, A^{Δ} ⊆ 𝔏_{v} and Θ = Θ_{[v,v]} is an invariant elation group.

(j) Only the possibility A^{Θ} ≠ A remains. The strategy in this case is to apply the dual (˜•) of (•) to the group Ω generated by α and the connected component of Δ_{u}. Assume first that A^{Ω} = A. Because dim(Θ_{u} ∪ Θ_{A}) < 12, there is an element ϑ ∈ Θ such that u^{ϑ} = z ≠ u and A^{ϑ} ≠ A. Remember that Θ ◁ Δ and that dim u^{Θ} = 4. Therefore, z^{Ω} ⊆ u^{Θ} and dim Ω_{z} ≥ 21. Let a ∈ A \ W, a ∉ A^{ϑ}. Then the connected component Λ of Ω_{a,z} satisfies dim Λ ≥ 13, moreover, Λ ≤ Ω^{ϑ} ≤ Δ_{A^{ϑ}}, and Λ fixes a non-degenerate quadrangle. Now (‡) implies Λ ≅ G_{2} . By assumption, Γ = Δ/Cs Θ is an irreducible subgroup of GL_{12}ℝ, and dim Γ ≤ 24 by Corollary 4. Since Λ ↪ Γ′, it follows from [18] (95.6) that Γ′ is almost simple and irreducible on Θ. Hence Γ′ ≅ Spin_{7}ℝ, but, according to the List, this group does not have an irreducible representation in dimension 12.

(k) Consequently, A^{Ω} ≠ A, and then dim Δ_{[u,u]} ≥ dim A^{Ω} > 0 by (˜•). If some oneparameter group in Δ_{[u,u]} has an affine axis, then, because of (f), all groups Δ_{[u,L]} with L ∈ 𝔏_{u} \ {W } have the same positive dimension, and the dual of [18] (61.12) implies Δ_{[u,W]} ≅ ℝ^{8} . Since u^{Δ} ≠ u, it follows that T is transitive. If each one-parameter subgroup of Δ_{[u,u]} has axis W however, then dim Δ_{[z,W]} > 0 for each center z ∈ u^{Δ} . Hence dim T > 4, and T contains a normal subgroup Θ̃ as claimed.

Theorem A is now an immediate consequence of the propositions.

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Received November 15, 1999