Lagrange’s Theorem for Moufang loops

S.M. Gagola III J.I. Hall Department of Mathematics Michigan State University East Lansing, Michigan 48824, U.S.A.

Submitted: 12 October 2004 Revised: 31 January 2005

Abstract

All finite Moufang loops have the Lagrange property.

1 Introduction

A finite loop L is said to have the Lagrange property if, for every subloop K of L, the order of K, written |K|, is a divisor of |L|, the order of L. Lagrange’s Theorem states that all finite groups have the Lagrange property. Here we resolve, in the positive, the long-standing conjecture that, more generally, all finite Moufang loops have the Lagrange property:

( 1.1) Theorem. Let K be a subloop of the finite Moufang loop L. Then |K| divides |L|.

It is well-known that the Lagrange property is valid for all Moufang loops if and only if it is valid for finite simple Moufang loops [4, Lemma V.2.1]. Of course, Lagrange’s Theorem holds if the simple loop is a group; so only nonassociative, finite, simple Moufang loops need be considered. These were classified by Liebeck [15], the only examples being the loops P (q) first studied by Paige [19]. The Paige loop P (q) is the central quotient of the loop of norm 1 units in Oct(q), the split octonians over the finite field 𝔽_{q}. Liebeck’s proof used Doro’s observation [7] that Moufang loops correspond to groups with triality. The corresponding triality group for P (q) is P Ω_{8}^{+}(q):Sym(3).

In our proof we show that it is enough to check divisibility for K maximal in L = P (q) and that these subloops correspond to maximal triality subgroups of P Ω_{8}^{+}(q):Sym(3). All maximal subgroups of P Ω_{8}^{+}(q):Sym(3) have been catalogued by Kleidman [14], and we show that each corresponding subloop conforms to the Lagrange property.

Primary subject: 20N05, Secondary subject: 20D06; Keywords: Moufang loop, Lagrange’s Theorem, group with triality

Earlier work on the Lagrange property for Moufang loops can be found in [5, 16, 24].

This paper was first submitted for publication in June 2004. We quickly learned that the paper [11] with the same result (and title!) had been written by A.N. Grishkov and A.V. Zavarnitsine and submitted in late 2003. The same authors have a second preprint [12] which finds the maximal subloops of the finite Paige loops (from which the Lagrange property follows, as mentioned above). More recently, we learned that Eric Moorhouse [17] has also independently classified the maximal subloops of the finite Paige loops and thereby verified the Lagrange property. The work of [11, 12, 17] depends, as does ours, on Kleidman’s classification [14] of the maximal subgroups of P Ω_{8}^{+}(q):Sym(3). The details of the several proofs are, however, different; and we have been encouraged by Doctors Grishkov and Zavarnitsine and by others to continue with publication of our version.

The first author’s thesis [9], prepared without prior knowledge of [11, 12, 17], improves upon the work of this paper and those in two ways. All finite subloops of arbitrary (possibly infinite) Paige loops are described, the classification of maximal subloops of the finite loops and the Lagrange property thus being corollaries. Since the containing loop need not be finite, Kleidman’s results [14] are not directly applicable. Furthermore Kleidman’s proof makes broad use of the classification of finite simple groups, and it is desirable to minimize reliance upon that major result. The results of [9] (which will appear elsewhere) are almost entirely elementary in proof, with only a few specific results from the classification used.

For general reference on loop theory, see [4, 20]. For group theory see [1], and for geometry see [21]. If s is an element of GL_{K}(V), then, for the vector v in V, the commutator [v, s] is defined to be v(s − 1) = −v + v.s. Similarly, for a subspace U of V and a subset S of GL_{K}(V), the commutator subspace [U, S] is the K-span of the various [u, s] with u ∈ U and s ∈ S.

2 Loops, designs, and triality

We introduce Moufang loops, Latin square designs, and groups with triality. These are linked in Theorem 2.2 (from [13]), which is fundamental to this paper. We also present some general results about groups with triality, especially those coming from subloops of Moufang loops.

A loop is a set L equipped with a binary product that possesses an identity element 1 and such that the equations ax = c and yb = c have unique solutions x and y for all a, b, c ∈ L. Thus a loop can be thought of as a group that might not be associative. A Moufang loop satisfies the weak associative law

((ab)a)c = a(b(ac)) ,

for all a, b, c ∈ L. Such loops were first studied by Moufang [18] in the context of alternative algebras. Clearly every group is a Moufang loop.

A Latin square design D is a pair D = (P, A) of points P = P(D) and lines A = A(D) (subsets of P) with the properties:

(i) P is the disjoint union of three parts R, C, E.

(ii) every line l ∈ A contains exactly three points, meeting each of R, C, E exactly once.

(iii) any pair of points from different parts belong to exactly one line.

We have |R| = |C| = |E| = n, say. The n^{2} lines of the Latin square design with |P | = 3n correspond to the n^{2} cells of a Latin square with side n. The line {a_{R}, b_{C}, c_{E}} asserts that in row a, column b of the Latin square one finds the entry c.

Let D = (P, A) be a Latin square design. Choose a line l_{0} ∈ A and a set L that we use to label R (one-to-one) in such a way that the point l_{0} ∩ R is labelled by the element 1 (or sometimes, for clarity, 1_{R}). We also will label C and E by L and, in doing so, define a multiplication on L with identity element 1. We call L the Thomsen loop of D, L = L(D). For a line l of A, we write [ a, b, c] in place of {a_{R}, b_{C}, c_{E}}.

The loop and labeling are given as follows:

1. Label l_{0} as [ 1, 1, 1 ], and label the points of R \ 1_{R} with L \ 1.

2. Label E with the set L^{−1} of right inverses of members of L via [ a, 1, a^{−1} ] ∈ A.

3. Label C with the members of L via [ 1, a, a^{−1} ] ∈ A.

4. The right inverse map is determined by [ a, a^{−1}, 1 ] = [ a, b, 1 ] ∈ A; so E, previously only labelled by L^{−1}, is now also labelled by L.

5. Define ab = c in L by [ a, b, c^{−1} ] ∈ A. (That is, [ d, e, f ] ∈ A if and only if (de)f = 1 in L.)

Choice of a different identity line l_{0} and of a different identification of L with R correspond to loop isotopy for L [4, 20].

Conversely, if L is a loop, then the Thomsen design of L, D = T(L), has point set P(L) = L_{R} ∪ L_{C} ∪ L_{E} of size 3|L| and line set A(L) consisting of those triples {a_{R}, b_{C}, c_{E}} = [ a, b, c] with (ab)c = 1 in L. (Thomsen [23] seems to have been the first to associate discrete incidence systems with arbitrary loops. Most of the earlier work on webs concentrated on Euclidean geometry.)

The two constructions are easily seen to be inverses of each other in the sense that T(L(D)) is isomorphic to D and L(T(L)) is isotopic to L.

In Theorem 2.2 below, we will see that Moufang loops correspond to Latin square designs admitting certain types of automorphisms. Here the automorphism group Aut(D) of the Latin square design D = (P, A) consists of all permutations g of the point set P that take lines to lines; that is, {a, b, c} ∈ A if and only if {ag, bg, cg} ∈ A.

For x ∈ P, say x ∈ R, consider the partial permutation τ_{x} that exchanges the sets C and E via

τ_{x}(y) = z, τ_{x}(z) = y if and only if [ x, y, z] ∈ A .

The question is whether τ_{x} can be extended to a permutation on all P (by defining it on R) so that τ_{x} is in Aut(D).

If τ_{x} has two extensions α and β to Aut(D), then αβ and αβ^{−1} are trivial on C ∪ E and so on D. That is, α = β^{−1} = β. Therefore, τ_{x} has at most one such extension, and if it exists then it has order 2. Thus we may, without confusion, call this extension τ_{x} as well. It is a central automorphism of D with center x.

Completely similar remarks are valid for central automorphisms τ_{y} and τ_{z} with centers y ∈ C and z ∈ E. If we want to emphasize the part of P from which the center comes, then we may write τ_{x} = ρ_{x}, τ_{y} = κ_{y}, and τ_{z} = ϵ_{z}.

( 2.1) Proposition. ([13, Prop. 2.3]) For an arbitrary Latin square design D = (P, A),

{τ_{p} | p ∈ P, τ_{p} ∈ Aut(D)}

is a normal set of elements of order 2 in Aut(D). For g, τ_{p} ∈ Aut(D), we have τ_{p}^{g} = τ_{pg}.

If τ_{p}, τ_{q} ∈ Aut(D) with p and q belonging to different parts of P, then τ_{p}τ_{q} has order 3. In this case {τ_{p} | p ∈ P, τ_{p} ∈ Aut(D)} is a single conjugacy class of Aut(D).

The case where τ_{p} ∈ Aut(D), for every point p of P, will lead to Moufang loops. The proposition suggests an abstract setting for such groups.

Let D = D^{G} be a normal set of elements of order 2 in the group G = ⟨D⟩. Then (G, D, π), or just G, is said to be a group with triality (or triality group) with respect to the set D of triality reflections (or just reflections) and projection π provided:

π is a homomorphism from G onto Sym(3), and |de| = 3 whenever d, e ∈ D with π(d) ≠ π(e).

That is, if d, e ∈ D with π(d) ≠ π(e) then ⟨d, e⟩ ≃ Sym(3). In particular d and e are conjugate in the group they generate, so the normal set D is actually a single conjugacy class of G. It is often (but not always) the case that the class D and projection map π are uniquely determined within the group G with triality.

The group ker(π) is called the rotation subgroup of (G, D, π).

Proposition 2.1 allows the construction of groups with triality from Latin square designs that admit the central automorphism τ_{p} for each point p. In this case we write D(D) = { τ_{p} | p ∈ P } and G(D) = ⟨D(D)⟩. By Proposition 2.1, the group G(D) is a group with triality whose reflection class is D(D). The projection map π: G(D) ⟶ Sym(ρ, κ, ϵ) is then given by

ρ_{p} ↦ (ρ)(κ, ϵ) , κ_{p} ↦ (κ)(ρ, ϵ) , ϵ_{p} ↦ (ϵ)(ρ, κ) ,

for p ∈ P. If L is a loop with D = T(L), then we write G(L) for G(D) and D(L) for D(D). Given a ∈ L, we also write ρ_{a} for τ_{a_{R}}, κ_{a} for τ_{a_{C}}, and ϵ_{a} for τ_{a_{E}}.

Conversely, given a group (G, D, π) with triality, we can define a Latin square design T((G, D, π)) with point set P = D. For each d, e ∈ D with π(d) ≠ π(e), we have ⟨d, e⟩ ≃ Sym(3); so we take the line of T((G, D, π)) on d, e to be {d, e, ded = ede}. We thus also refer to the subgroup ⟨d, e⟩ ≃ Sym(3) as a line of the group (G, D, π) with triality.

Usually we can write T(G) for T((G, D, π)) without ambiguity. If L is a loop with T((G, D, π)) = T(L), then we also write L = L((G, D, π)) = L(G). This is an abuse, since L is only determined up to isotopy by (G, D, π).

The fundamental connection between arbitrary Moufang loops and abstract groups with triality was observed and studied by Doro [7], motivated by the work of Glauberman [10]. Doro’s definition of triality is different but equivalent to the one given here. In Doro’s treatment the rotation subgroup ker(π), rather than the larger group G, is called a group with triality.

Bol [3] gave a geometric characterization of 3-nets or 3-webs (dual to Latin square designs—see [4, I.4] or [20, II.1]) that are coordinatized by Moufang loops. Tits [22] showed that Bol’s 3-nets admit reflections as automorphisms. (See also Bruck [4, p.120].) Tits also noted and discussed the connection with triality for orthogonal groups and the octonians. Our treatment follows that of [13], which was in part motivated by the work of Funk and Nagy [8].

( 2.2) Theorem. ([13, Theorem 3.6]) The following are equivalent: (1) L is a Moufang loop;

(2) D = (P, A) is a Latin square design admitting a central automorphism at each point;

(3) (G, D, π) is a group with triality and Z(G) = 1. Indeed:

given L as in (1), we have D = T(L), G = G(L), and D = D(L); given D as in (2), we have L = L(D), G = G(D), and D = D(D); given G as in (3), we have L = L(G) and D = T(G).

Under (1), the map π = π_{L} is determined by projection onto ⟨ρ_{1}, κ_{1}, ϵ_{1}⟩ ≃ Sym(3). That is, ρ_{a} ↦ ρ_{1}, κ_{a} ↦ κ_{1}, and ϵ_{a} ↦ ϵ_{1}, for a ∈ L. Similarly, under (2), a suitable map π = π_{D} can be determined by projection onto any line.

In this correspondence, we have

3|L| = |P | = |D| .

These equalities are crucial for our study of the Lagrange property.

If (G, D, π) is a group with triality and Q is a normal subgroup of G that is contained in ker(π), then G/Q is also a group with triality. Subgroups of G also give rise to new groups with triality. We say that the subgroup H of G admits the triality if (⟨D ∩ H⟩, D ∩ H, π|_{⟨D∩H⟩}) is itself a group with triality. The following is elementary:

( 2.3) Lemma. Let (G, D, π) be a group with triality, and let H be a subgroup of G. Then the following are equivalent:

(1) H admits the triality;

(2) H contains a line;

(3) G = ker(π).H and H ∩ D is nonempty.

Certain properties of groups with triality, particularly those regarding cardinality, can be discussed in terms of quotients and subgroups.

( 2.4) Lemma. Let (G, D, π) be a group with triality. Set M = ker(π), the rotation subgroup.

(1) Assume that the subgroup H of G admits the triality, and let M_{0} be the rotation subgroup of H_{0} = ⟨D ∩ H⟩. The set D ∩ H = D ∩ H_{0} is a single conjugacy class of H_{0}. For r ∈ D ∩ H, we have

|D ∩ H| = 3|D ∩ rM_{0}| and D ∩ rM_{0} = r^{M_{0}} = r^{M∩H} .

(2) Assume Q is a normal subgroup of G that is contained in M, and set Ḡ = G/Q. Let I be a line of G with r ∈ D ∩ I. Then Q.I admits the triality and (Ḡ, D̄, π̄) is a group with triality. Furthermore

|D| = |D̄| · |D ∩ rQ| = 3|r̄M̄| · |r^{Q}| .

Proof. (1) By definition and Lemma 2.3, H_{0} is a group with triality. As already seen in Proposition 2.1 and the remarks that follow it, D ∩ H = D ∩ H_{0} is a single class of H_{0}. These involutions are uniformly distributed among the cosets corresponding to the involutions of π(H_{0}) ≃ Sym(3), so |D ∩ H| = |D ∩ H_{0}| = 3|D ∩ rM_{0}|.

Clearly D ∩ rM_{0} ⊇ r^{M∩H} ⊇ r^{M_{0}}. In H_{0}/M_{0} ≃ Sym(3), we find ⟨r⟩M_{0} = N_{H_{0}}(rM_{0}). Therefore

D ∩ rM_{0} = r^{H_{0}} ∩ rM_{0} = r^{⟨r⟩M_{0}} = r^{M_{0}} ,

completing (1).

(2) This is immediate except for the last line. There |D̄| = 3|D̄∩r̄M̄| = 3|r̄M̄| and D ∩ rQ = r^{Q}, both by (1).

( 2.5) Proposition. Let D = (P, A) be a Latin square design. Consider the group Univ(D) with generators d̃, one for each d ∈ P, and relations d̃^{2} = 1 and d̃ẽd̃ = f̃ = ẽd̃ẽ, for all d ∈ P and [ d, e, f ] ∈ A. Then Univ(D) is a triality group with reflection class D̃ = { d̃| d ∈ P } whose projection map π̃ with image Sym(ρ, κ, ϵ) is given by

r̃ ↦ (ρ)(κ, ϵ) , c̃ ↦ (κ)(ρ, ϵ) , ẽ ↦ (ϵ)(ρ, κ) ,

for all r ∈ R, c ∈ C, and e ∈ E.

Assume additionally that the Latin square design D = (P, A) admits a central automorphism at each point. Then we have T(Univ(D)) ≃ D and Univ(D)/Z(Univ(D)) ≃ G(D) with D̃ mapped bijectively to D(D). Indeed any group with triality whose associated Latin square design is isomorphic to D is a central quotient of Univ(D).

Proof. By construction, for an arbitrary Latin square design D, D̃ is a conjugacy class of elements of order 2 in G̃ = Univ(D) and π̃ is a homomorphism onto Sym(ρ, κ, ϵ). Furthermore, if x, y ∈ D̃ with π̃(x) ≠ π̃(y) then |xy| = 3; so Univ(D) is a group with triality as described.

For most Latin square designs, Univ(D) will itself be symmetric of degree 3. If, however, D admits a central automorphism at each point, then d̃ ↦ τ_{d}, for all d ∈ P, is a bijection of D̃ and D(D); and it describes a homomorphism of Univ(D) onto G(D) since all relations are determined by D. If Z is the kernel of this homomorphism, then d̃ = D̃ ∩ d̃Z, for all d ∈ P. Therefore d̃^{Z} = d̃, and Z is central in Univ(D). As G(D) has trivial center, in fact Z = Z(Univ(D)).

Indeed, if (G, D, π) is a group with triality and T((G, D, π)) ≃ D, then the argument of the previous paragraph remains valid except we no longer know that G has trivial center. Still we find that G is isomorphic to Univ(D)/Z_{0} for some Z_{0} ≤ Z(Univ(D)).

The group Univ(D) is called the universal triality group of D, and we also write Univ(L) = Univ(D) for any Moufang loop L with D = T(L). The proposition should be compared with Doro’s [7, Theorem 2], which provides a presentation for the rotation group ker(π̃) of Univ(L).

Let L be a Moufang loop. If K is a subloop of L, then we set D_{L}(K) = { ρ_{a}, κ_{a}, ϵ_{a} | a ∈ K } of size 3|K| and G_{L}(K) = ⟨D_{L}(K)⟩. Thus G(L) = G_{L}(L) and D(L) = D_{L}(L). Always G_{L}(K) contains the identity line I(L) = ⟨ρ_{1}, κ_{1}, ϵ_{1}⟩ ≃ Sym(3) and so admits the triality.

( 2.6) Proposition. Let L be a Moufang loop. Set G = G(L), D = D(L), and π = π_{L}.

(1) Let K be a subloop of L. Then H = G_{L}(K) is a triality subgroup of G with reflection class D_{L}(K) = D ∩ H = D ∩ N_{G}(H) and projection map equal to the restriction π|_{H}. We have |D ∩ H| = 3|K|, and H is a central quotient of Univ(K) as group with triality.

(2) If I(L) ≤ N ≤ G, then there is a subloop K of L with D_{L}(K) = D ∩ N and G_{L}(K) = ⟨D ∩ N⟩.

(3) If K is a maximal subloop of L, then N_{G}(G_{L}(K)) is a maximal subgroup of G with G = ker(π).N_{G}(G_{L}(K)).

Proof. (1) As G_{L}(K) contains I(L), it admits the triality by Lemma 2.3. By Proposition 2.5, G_{L}(K) is a central quotient of Univ(K) and |D ∩ H| = |D_{L}(K)| = 3|K|. By Lemma 2.4.1 applied to N_{G}(H), the set D ∩ N_{G}(H) is a single conjugacy class of N_{G}(H) and so must be the class D ∩ H.

(2) Let M = ker(π). We know that ρ_{1} acts on C = D∩κ_{1}M and E = D∩ϵ_{1}M via

a_{C}.ρ_{1} = (a^{−1})_{E} and a_{E}.ρ_{1} = (a^{−1})_{C} ,

for all a ∈ L. To describe its action on R = D ∩ ρ_{1}M, consider l = [ a, a^{−1}, 1 ] = {a_{R}, (a^{−1})_{C}, 1_{E}}, a line of A(L). Thus

l.ρ_{1} = {a_{R}.ρ_{1}, (a^{−1})_{C}.ρ_{1}, 1_{E}.ρ_{1}} = {a_{R}.ρ_{1}, a_{E}, 1_{C}} = [ a.ρ_{1}, 1, a]

is also in A(L). The unique line of shape [ ∗, 1, a] is [ a^{−1}, 1, a], so we conclude that a_{R}.ρ_{1} = (a^{−1})_{R}. Similar statements hold for κ_{1} and ϵ_{1}, giving

a_{R}.ρ_{1} = (a^{−1})_{R} , a_{C}.κ_{1} = (a^{−1})_{C} , a_{E}.ϵ_{1} = (a^{−1})_{E} ,

for all a ∈ L. This in turn allows us to calculate the action of t = κ_{1}ρ_{1} on D, and we find t = (R, C, E) ; that is,

a_{R}.t = a_{C} , a_{C}.t = a_{E} , a_{E}.t = a_{R} ,

for all a ∈ L.

Let K = { a ∈ L | ρ_{a} ∈ N }. As t ∈ I(L) ≤ N, we have also K = { a ∈ L | κ_{a} ∈ N } = { a ∈ L | ϵ_{a} ∈ N }. As ρ_{1} ∈ I(L) ≤ N, the subset K of L contains 1 and is closed under inverses. If a, b ∈ K then ρ_{a}, κ_{b} ∈ N and ρ_{a}κ_{b}ρ_{a} = ϵ_{c} ∈ N, thus c = (ab)^{−1} ∈ K and ab ∈ K. Therefore K is a subloop of L; so we have verified D ∩ N = D_{L}(K) and ⟨D ∩ N⟩ = G_{L}(K), as claimed.

(3) As N = N_{G}(G_{L}(K)) = N_{G}(D_{L}(K)) (by (1)) contains the line I(L), we have G = ker(π).N. Suppose N is not maximal in G, and choose n ∈ G\N with N^{∗} = ⟨n, N⟩ proper in G. As n ∉ N, we have (D∩N)^{n} ≠ D∩N = D_{L}(K). Thus D ⊃ D∩N^{∗} ⊃ D∩N. By (2), there is a subloop K^{∗} of L with D(K^{∗}) = D∩N^{∗} and L > K^{∗} > K. Therefore K is not maximal in L.

3 Paige loops and their groups

The fundamental group theoretic result in the classification of finite, nonassociative, simple Moufang loops was the following:

( 3.1) Theorem. (Doro [7], Liebeck [15]) Let (G, D, π) be a finite group with triality, and let I be a line of G. Let M be a semisimple, I-invariant subgroup of G, and let E be a nonabelian simple direct factor of M. Then we have one of:

(1) [E, I] = 1;

(2) ⟨E, I⟩ ≃ E^{3}:I, a wreathed product;

(3) ⟨E, I⟩ = E:I ≃ P Ω_{8}^{+}(q):Sym(3), for some q, and I is uniquely determined within E:I up to conjugacy.

Proof. The transitive I-space Ω = {E^{I}} has size dividing 6. As ⟨Ω⟩.I admits the triality, |Ω| is not 2 or 6 by [7, Prop. 1]. In the same place, Doro showed that |Ω| = 3 gives the wreathed case (2).

If |Ω| = 1, then either [I, E] = 1 or I induces outer automorphisms on E by [7, Cor. 4, p.382]. In the latter case, Liebeck [15] showed we must have (3).

Doro [7] also showed that, under Theorem 3.1.2, the loop L(⟨E, I⟩) is isomorphic to the group E. Therefore our verification of the Lagrange property involves a careful study of the group and associated loop occuring under Theorem 3.1.3.

Over the finite field 𝔽_{q} a nondegenerate 8-dimensional composition algebra is uniquely determined up to isomorphism as the 𝔽_{q}-algebra of split octonians Oct(q) ([19, §2]). These can be conveniently written as Zorn’s vector matrices

m = [ \begin{matrix}a & b⃗ \\ c⃗ & d \end{matrix} ]

with a, d ∈ 𝔽_{q} and b⃗, c⃗ ∈ 𝔽_{q}^{3}. Multiplication is given by

[ \begin{matrix}a & b⃗ \\ c⃗ & d \end{matrix} ] [ \begin{matrix}x & y⃗ \\ z⃗ & w \end{matrix} ] = [ \begin{matrix}ax + b⃗ · z⃗ & ay⃗ + wb⃗ − c⃗ × z⃗ \\ xc⃗ + dz⃗ + b⃗ × y⃗ & c⃗ · y⃗ + dw \end{matrix} ]

using the standard dot and cross products of 3-vectors. The associated quadratic form, which admits composition, is the norm (or determinant) Δ(m) = ad−b⃗·c⃗. (See [15] for a different rendering of Oct(q).)

In Oct(q) an element m is invertible if and only if Δ(m) ≠ 0, and the loop of units GLL(q) is a Moufang loop. This possesses a normal subloop SLL(q) consisting of all units with norm 1. The scalars of SLL(q) form a normal subloop {±I} of order d = gcd(q − 1, 2), and the Paige loop is the quotient P (q) = P SLL(q) = SLL(q)/{±I}. (See [19] for all of this.) The Zorn construction of the split octonian algebra Oct(F) and its Moufang loop of units GLL(F) can be made over arbitrary fields F. Also of interest is the quotient of the full unit loop by scalars: P GLL(F) = GLL(F)/{ αI | α ∈ F }. The Paige loop P (q) has index d in P GLL(q) as the squares of 𝔽_{q}^{∗} have index d in 𝔽_{q}^{∗}.

( 3.2) Theorem. (Paige [19]) The loop P (q) is a simple, nonassociative, Moufang loop. We have |P (q)| = q^{3}(q^{4} − 1)/d, where d = gcd(q − 1, 2).

( 3.3) Lemma. For prime power q, set d = gcd(q−1, 2) and ℓ(q) = q^{3}(q^{4}−1)/d. (1) If q_{0} divides q, then ℓ(q_{0}) divides ℓ(q).

(2) If odd q_{0}^{2} = q, then 2ℓ(q_{0}) divides ℓ(q).

(3) ℓ(2) = 120 divides ℓ(q), for all q.

Proof. This is elementary.

We have G(P (q)) = P Ω_{8}^{+}(q):Sym(3) as in Theorem 3.1.3. This is wellknown [7, 15] and a consequence of Theorems 3.1 and 3.2. An elementary proof, using only the Cartan-Dieudonné Theorem [21, Theorem 11.39], that G(P (F)) = P Ω_{8}^{+}(F):Sym(3), for all fields F, can easily be extracted from [2, Theorem 1]. Also evident from [2] is that a triality reflection of P Ω_{8}^{+}(F):Sym(3) induces a reflection on P O_{8}^{+}(F) (discussed further below) and that ker(π) is the corresponding rotation subgroup P Ω_{8}^{+}(F) (hence the terminology). The normal subgroup of Aut(P Ω_{8}^{+}(q)) generated by this reflection class is P Ω_{8}^{+}(q):Sym(3) when q is even but is P Ω_{8}^{+}(q):Sym(4) when q is odd. For odd q, within Aut(P Ω_{8}^{+}(q)) there are four triality subgroups P Ω_{8}^{+}(q):Sym(3), permuted faithfully by P Ω_{8}^{+}(q):Sym(4)/P Ω_{8}^{+}(q) ≃ Sym(4). See also [14, §1.4]. (More generally [2], the normal subgroup G^{∗} of Aut(P Ω_{8}^{+}(F)) generated by the reflection class has G^{∗}/P Ω_{8}^{+}(F) ≃ Q^{2}:Sym(3), where Q is the multiplicative group of the field F modulo its squares. Indeed G^{∗} = G(P GLL(F)), the triality group associated with the octonian loop of units modulo scalars.)

The related group O_{8}^{+}(q) acts as the full isometry group of the 8-dimensional orthogonal space (V, Δ), where V = 𝔽_{q}^{8} is the vector space carrying Oct(q). This orthogonal space has +-type; that is, it is the perpendicular direct sum of nondegenerate 2-spaces, each of which contains singular vectors. A vector v is singular if Δ(v) = 0, in which case the subspace spanned by v is also singular. A nonsingular 1-space is of +-type if all its Δ values are squares and otherwise is of −-type. (Thus a 1-space is of +-type if and only if it contains a vector v with Δ(v) = 1.) A nondegenerate subspace is of −-type if it is not of +-type, and a subspace is totally singular if all its vectors are singular.

The intersection of P O_{8}^{+}(q) with P Ω_{8}^{+}(q):Sym(3) is P Ω_{8}^{+}(q)⟨r⟩ where r represents a +-reflection r̂ of O_{8}^{+}(q). In saying that r̂ is a +-reflection we mean that, for its action on V, the center [V, r̂] = V (r̂ − 1) is a nonsingular 1-space of +-type (so in characteristic 2, the +-reflections are actually transvections).

Recall that a subspace W of V is invariant under the +-reflection s if and only if either W contains the center [V, s] or W is perpendicular to the center: W ≤ [V, s]^{⊥} = C_{V} (s).

In odd characteristic Aut(P Ω_{8}^{+}(q)) does not act on Ω_{8}^{+}(q) but instead on its double cover Spin_{8}^{+}(q). Nevertheless, we occasionally abuse terminology by discussing the action of elements and subgroups H of P Ω_{8}^{+}(q):Sym(3) on the underlying vector space V, for instance, blurring the distinction between a +-reflection r̂ of O_{8}^{+}(q) and the element r of P O_{8}^{+}(q) representing it. For H ≤ P O_{8}^{+}(q) we should more properly lift H to a subgroup Ĥ ≤ O_{8}^{+}(q), so that H ≃ Ĥ/Ẑ with Ẑ = Z(O_{8}^{+}(q)) of order gcd(q − 1, 2). Further care is required when H is not conjugate into P O_{8}^{+}(q) (that is, when 3 divides |π(H)|). In that case, we must confine our discussion to the way H acts on the set of subspaces of the associated D_{4} geometry—the set S^{1} of totally singular 1-spaces, the set S^{2} of totally singular 2-spaces, and the two classes S_{1}^{4} and S_{2}^{4} of totally singular 4-spaces of V. (Two totally singular 4-spaces W_{1} and W_{2} belong to the same class if and only if W_{1} ∩ W_{2} has even dimension.) The group P Ω_{8}^{+}(q):Sym(3) leaves S^{2} invariant and permutes the members of S^{1} ∪S_{1}^{4} ∪S_{2}^{4}, inducing Sym(3) on {S^{1}, S_{1}^{4}, S_{2}^{4}}. See [2, §§2-3].

We make extensive use of Kleidman’s work [14] on maximal subgroups of groups G for P Ω_{8}^{+}(q) ≤ G ≤ Aut(P Ω_{8}^{+}(q)). In particular, we use his notation for such maximal subgroups.

( 3.4) Theorem. Let N be a maximal subgroup of G = P Ω_{8}^{+}(q):Sym(3) with G = P Ω_{8}^{+}(q).N. Then N_{0} = N ∩ P Ω_{8}^{+}(q) has one of the types:

G_{2}, K_{3}, S_{a}, S_{2}, K_{5}, N_{1}, N_{2}, N_{3}, N_{4}, P_{2}, R_{s2}, I_{+2}, I_{−2}, I_{+4} .

Conversely, for each of these types, there is a unique P Ω_{8}^{+}(q)-conjugacy class of subgroups N_{0} with G = N_{G}(N_{0})P Ω_{8}^{+}(q).

Proof. By Kleidman’s results [14, Prop. 4.2.1, Table III], all such N_{0} must have one of these types or K_{4} or K_{6}. But subgroups N_{0} of those two types have P Ω_{8}^{+}(q)N_{G}(N_{0}) proper in G; see [14, 2.3.4,2.3.9].

Uniqueness follows immediately from [14, Table I], except when q is odd and N_{0} has type K_{3}, S_{2}, K_{5}, or N_{4}. In the exceptional cases P Ω_{8}^{+}(q) has four conjugacy classes of subgroups N_{0}, and these are permuted by Aut(P Ω_{8}^{+}(q)) and P Ω_{8}^{+}(q):Sym(4) with induced action Sym(4). Therefore exactly one of the classes satisfies G = P Ω_{8}^{+}(q)N_{G}(N_{0}).

In Table K below, we have extracted what is most important for us from Kleidman’s Tables I [14, pp.186–191] and III [14, p.238]. Table K lists the conjugacy classes of subgroups N_{0} of P Ω_{8}^{+}(q) for which N = N_{G}(N_{0}) is maximal in G = P Ω_{8}^{+}(q):Sym(3) and has P Ω_{8}^{+}(q).N = G. We call such an N a maximal complementing subgroup of P Ω_{8}^{+}(q):Sym(3). A complementing N might not contain reflections and so not admit the triality. (In fact only in the cases K_{3} and N_{3} does complementing N fail to admit the triality [9].)

The first column of the table lists Kleidman’s names for the cases. (We have surpressed Kleidman’s additional superscripts, since he must consider more than one class while we only have one.) The second column gives the isomorphism type of N_{0}. Here the symbol ↓ indicates that the subgroup described is actually the preimage in Ω_{8}^{+}(q), so a central subgroup of order d = gcd(q − 1, 2) must be factored out. The third column provides restrictions on the field 𝔽_{q} and characteristic p without which the corresponding N is not maximal. The notation \frac{1}{d}A indicates a subgroup of index d in A. The group A^{a} is the direct product of a copies of A; especially p^{a} is elementary abelian of that order. A group [q^{a}] has order q^{a} but undescribed isomorphism type.

Table K. Maximal complementing subgroups of P Ω_{8}^{+}(q):Sym(3)

Name  Isomorphism type of N_{0}  Restrictions
G_{2} G_{2}(q)  −
K_{3} P GL_{3}^{ϵ}(q)  ϵ = ±, 2 < q ≡ ϵ1 (mod 3)
S_{a} P Ω_{8}^{+}(q_{0})  q = q_{0}^{a}, a prime
S_{2} P Ω_{8}^{+}(q_{0}).2^{2}  q = q_{0}^{2}, odd
K_{5} P Ω_{8}^{+}(2)  q = p ≥ 3
N_{1} ↓(\frac{1}{d}Z_{q+1} × \frac{1}{d}GU_{3}(q)) .2^{d}  −
N_{2} ↓(\frac{1}{d}Z_{q−1} × \frac{1}{d}GL_{3}(q)) .2^{d}  q ≥ 4
N_{3} (D_{(2/d)(q^{2}+1)} × D_{(2/d)(q^{2}+1)}) .2^{2} q ≠ 3
N_{4} [2^{9}]:GL_{3}(2)  q = p ≥ 3
P_{2} ↓[q^{11}] (\frac{1}{d}Z_{q−1})^{2}:\frac{1}{d}GL_{2}(q).d^{2}  −
R_{s2} ↓[q^{9}]:(\frac{1}{d}GL_{2}(q) × Ω_{4}^{+}(q)).d  −
I_{+2} ↓(\frac{1}{d}Z_{q−1})^{4}.d^{3}.2^{3}.Sym(4)  q ≥ 7
I_{−2} ↓(\frac{1}{d}Z_{q+1})^{4}.d^{3}.2^{3}.Sym(4)  q ≠ 3
I_{+4} (Ω_{4}^{+}(q) ∘ Ω_{4}^{+}(q)) .2.2^{d}  q ≥ 3

4 Maximal triality subgroups of P Ω_{8}^{+}(q):Sym(3)

( 4.1) Theorem. Let G = P Ω_{8}^{+}(q):Sym(3) with D its class of triality reflections. If N is a maximal subgroup of G with G = P Ω_{8}^{+}(q).⟨D ∩N⟩, then |D ∩N| divides |D|.

Given this theorem, we have

Proof of Theorem 1.1:

Let L be a finite Moufang loop and K a subloop. The proof is by induction on |L|, the result being clear for |L| = 1.

By [4, Lemma V.2.1] and induction, L is simple. (Lemma 2.4.2 could also be used in place of the reference.) As Lagrange’s Theorem holds in groups, we may assume that L is nonassociative. Therefore by Liebeck’s theorem [15], L is P (q) for some prime power q.

The result is evident if L = K, so we may assume that K is proper in L and let K_{1} be a maximal subloop of L containing K. By induction |K| divides |K_{1}|, so we are reduced to the case in which K_{1} = K is a maximal subloop of L.

Let N be the normalizer of G_{L}(K) in G = G(L) = P Ω_{8}^{+}(q):Sym(3). By Proposition 2.6.3, N is a maximal subgroup of G with G = P Ω_{8}^{+}(q).N. By Proposition 2.6.1 and Theorem 4.1, 3|K| = |D∩N| divides |D| = 3|L|. Therefore |K| divides |L|, as desired.

It remains to prove Theorem 4.1. Let G = P Ω_{8}^{+}(q):Sym(3) with D its class of triality reflections. By Theorem 3.1.3, the class D is uniquely determined as is the line class I^{G}. The projection π is also unique, given by π: G ⟶ G/G′′ ≃ Sym(3). We let M be the rotation subgroup ker(π) = G′′ = P Ω_{8}^{+}(q). Fix a line I, and let O_{3}(I) = ⟨t⟩ and r = D ∩ I ∩ P O_{8}^{+}(q). Set d = gcd(q − 1, 2). By Proposition 2.6 and Theorem 3.2 we have |D| = 3|P (q)| = 3ℓ(q) = 3q^{3}(q^{4}−1)/d.

Let N be maximal in G as in Theorem 4.1. Therefore N is a maximal complementing subgroup in G, and the group N_{0} = N ∩ M has one of the types on Kleidman’s list as given in Theorem 3.4 and Table K. We treat the various cases in a sequence of propositions. By Theorem 3.1.3 we may assume I ≤ N, so N = N_{0}:I.

The involution r of I is a +-reflection of P O_{8}^{+}(q). It is important that, by Lemma 2.4.1, the set D ∩ rN_{0} = r^{N_{0}} consists of all the +-reflections of P O_{8}^{+}(q) that are contained in N. Therefore when N and N_{0} have natural geometric descriptions in terms of V = 𝔽_{q}^{8}, the number |D ∩ N| = 3|D ∩ rN_{0}| can be calculated by counting +-reflections or their associated centers.

( 4.2) Lemma. Let H_{0} be a normal subgroup of N_{0} with C_{N_{0}}(H_{0}) = 1. Then either [H_{0}, I] ≠ 1 or [N_{0}, I] = 1.

Proof. If [H_{0}, I] = 1, then I = C_{N}(H_{0}) is normal in N = N_{0}:I. Therefore [N_{0}, I] ≤ N_{0} ∩ I = 1.

( 4.3) Proposition. (1) The group N_{0} has type G_{2} if and only if N = N_{0} ×I. In this case N = N_{G}(⟨t⟩) = N_{G}(I) ≃ G_{2}(q) × Sym(3) and |D ∩ N| = 3.

(2) The group N_{0} does not have type K_{3}.

Proof. For N_{0} of type G_{2} or K_{3}, the derived group H_{0} = N′_{0} is a simple group not isomorphic to any P Ω_{8}^{+}(q_{0}) and has trivial centralizer in N_{0}.

The subgroup H_{0}.I admits the triality, so by Theorem 3.1 we must have [H_{0}, I] = 1. By Lemma 4.2, we have [N_{0}, I] = 1. (Here and elsewhere we use Theorem 3.1 in the form: if I normalizes B ≤ M and I does not centralize a nonsolvable B-composition factor, then the isomorphism type of that factor either is P Ω_{8}^{+}(q_{0}), for some q_{0}, or appears as an B-composition factor with multiplicity at least three.)

In (1), N_{0} ≃ G_{2}(q) is indeed centralized by I and is also the centralizer in M of the 3-element t. (See [14, Prop. 3.1.1].)

In (2), N_{0} is P GL_{3}(q) (= P GL_{3}^{+}(q)) or P GU_{3}(q) (= P GL_{3}^{−}(q)). We have N = N_{0}.I < N_{G}(I) of type G_{2}, against maximality.

In view of Proposition 4.3 we may now assume that N_{0} is not contained in C_{G}(I) = C_{M}(t) = N_{M}(⟨t⟩) = N_{M}(I). Indeed, we may assume that I centralizes no appropriate H_{0} as in Lemma 4.2.

( 4.4) Proposition. If N_{0} has type S_{a}, S_{2}, or K_{5}, then |D∩N| is, respectively, 3ℓ(q_{0}) (for q = q_{0}^{a}), 6ℓ(q_{0}) (for q = q_{0}^{2}), or 3ℓ(2) and divides |D| = 3ℓ(q).

Proof. For S_{a} and K_{5}, the subgroup N_{0} is P Ω_{8}^{+}(q_{0}) with, respectively, q = q_{0}^{a} and q_{0} = 2. As I acts nontrivially on N_{0}, by Theorem 3.1 we have ⟨D ∩ N⟩ = P Ω_{8}^{+}(q_{0}):Sym(3). In this case |D ∩ N| = 3ℓ(q_{0}) by Theorems 3.1.3 and 3.2. Therefore |D ∩N| divides |D| = 3ℓ(q) by Theorem 3.2 and Lemma 3.3.

For S_{2}, N_{0} is P Ω_{8}^{+}(q_{0}).2^{2} with q = q_{0}^{2}, odd. By Lemma 4.2, I acts nontrivially on N′_{0} = P Ω_{8}^{+}(q), hence by Theorem 3.1 we have N′_{0}.I = P Ω_{8}^{+}(q_{0}):Sym(3). Therefore N_{0}.I = N ≃ P Ω_{8}^{+}(q_{0}):Sym(4) ≤ Aut(P Ω_{8}^{+}(q_{0})). Hence |D ∩ N| = 6ℓ(q_{0}), a divisor of |D| = 3ℓ(q_{0}^{2}) by Theorem 3.2 and Lemma 3.3.2.

Remark. Indeed G does contain subgroups with triality of each of these types [9].

For N_{0} of type S_{a}, we have P Ω_{8}^{+}(q_{0}):Sym(3) with q_{0}^{a} = q, corresponding to the subfield subloop P (q_{0}) of P (q). For N_{0} of type S_{2}, we have G_{L}(K) = P Ω_{8}^{+}(q_{0}):Sym(4), corresponding within P (q_{0}^{2}) to the subloop K = P GLL(q_{0}) of all unit octonians over 𝔽_{q_{0}} (not just those of norm 1) modulo scalars, since the nonsquares of 𝔽_{q_{0}} are squares in 𝔽_{q_{0}^{2}}.

The unit integral octonians span (modulo {±I}) a subloop P (2) [6]. Therefore all Paige loops P (F) contain P (2), as in K_{5}.

( 4.5) Proposition. If N_{0} has type N_{1} or N_{2}, then |D ∩ N| is, respectively, 3(q + 1)/d or 3(q − 1)/d and so divides |D|.

Proof. By Lemma 2.4, |D ∩ N| = 3|D ∩ rN_{0}|. Here D ∩ rN_{0} = r^{N_{0}} consists of those +-reflections of P O_{8}^{+}(q) that are in N.

See [14, Prop. 3.2.2-3]. Set N_{−} = N_{1} and N_{+} = N_{2}. Then N_{0} of type N_{ϵ} leaves invariant a decomposition V = U_{ϵ} ⊥ W_{ϵ}, where U_{ϵ} is a nondegenerate 2-space of ϵ-type. On U_{ϵ} the group N_{0} induces at least Ω_{2}^{ϵ}(q) ≃ Z_{q−ϵ1}, transitive on the nonsingular 1-spaces of +-type in U_{ϵ}. If (q, ϵ) = (2, −), set E = N′_{0} and otherwise set E = N′′_{0}. Then the nondegenerate 6-space W_{ϵ} = [V, E] of ϵ-type admits the characteristic subgroup E ≃ SL_{3}^{ϵ}(q) of N_{0} acting irreducibly. (Here SL_{3}^{+}(q) = SL_{3}(q) and SL_{3}^{−}(q) = SU_{3}(q).) Elements of rN_{0} must therefore respect the decomposition.

If (q, ϵ) ≠ (2, −) then E is quasisimple, and [I, E] = 1 by Theorem 3.1.3. When (q, ϵ) = (2, −), the characteristic subgroup Z(E) ≃ Z_{3} is fixed-point-free on W_{−}; so the +-reflection r does not invert Z(E) and instead must centralize it. Therefore in all cases no +-reflection of r^{N_{0}} has its center in W_{ϵ}.

A +-reflection s of r^{N_{0}} must leave W_{ϵ} invariant, so by the previous paragraph its center belongs to W_{ϵ}^{⊥} = U_{ϵ}. Conversely, the +-reflection r has its center in U_{ϵ}, so r^{N_{0}} contains all +-reflections with center in U_{ϵ}. Thus we have |D ∩ N| = 3|D ∩ rN_{0}| = 3(q − ϵ1)/d, always a divisor of 3ℓ(q) = 3q^{3}(q^{4} − 1)/d.

Remark. Here we find subloops of order (q−ϵ1)/d that are cyclic subgroups of P (q) [9]. Especially they are not maximal, since Moufang proved that all 2generated Moufang loops are groups ([4, §7.4],[20, IV.2.1]).

( 4.6) Proposition. N_{0} does not have type N_{3}.

Proof. A subgroup N_{0} of type N_{3} is the normalizer of a Sylow l-subgroup, for any odd prime divisor l of q^{2} + 1. (See [14, Prop. 3.3.1] for properties of subgroups of type N_{3}.) N_{0} leaves invariant a direct sum decomposition into 4-spaces, V = W_{0} ⊥ W_{1}. It contains a characteristic subgroup Z ≃ Z_{l} × Z_{l} that acts on each W_{i} irreducibly without fixed points, the two representations not being isomorphic. An element g ∈ O_{8}^{+}(q) of order 2 that switches W_{0} and W_{1} has [V, g] of dimension 4 and so is not a reflection. A +-reflection s of N would therefore fix each W_{i}, acting on Z and the induced Z_{l}. But then, for 1 ≠ z ∈ Z, the dihedral group S = ⟨s, s^{z}⟩ of order 2l would be irreducible on [W_{i}, S] = W_{i}, a 4-space, for one of the values of i. This is impossible since S is generated by two reflections.

( 4.7) Proposition. If N_{0} has type N_{4}, then |D∩N| divides 24 and so divides |D|.

Proof. Here N_{0} ≃ [2^{9}]:GL_{3}(2), where S = O_{2}(N_{0}) of order 2^{9} has S′ = Z(S) ≃ 2^{3} with GL_{3}(2) acting naturally. (In [14] see 3.4.2 and the proof of 4.1.10.)

By Theorem 3.1.3 we have [I, N_{0}] ≤ S. Let H ≤ N_{0} with H ≃ GL_{3}(2). The 3-element t of I must have a fixed point in its action on the N-space {H^{N_{0}}} = {H^{S}} of size a power of 2. Therefore we may assume t ∈ N_{G}(H). But then [t, H] ≤ H ∩ S = 1, so H ≤ C_{M}(t) = C_{M}(I) by Proposition 4.3.1.

The group I × H acts on Z(S) with H irreducible, so [I, Z(S)] = 1. With N̄ = N/Z(S), we then have by the previous paragraph and Lemma 2.4

|D ∩ N| = |D ∩ IS| = |D̄ ∩ \overline{IS}| = 3|D̄ ∩ r̄S̄| .

As S̄ is abelian of order 2^{6}, we find that |D ∩ N| divides 3.2^{3}. By Lemma 3.3.3, this divides |D|.

Remark. Maximal subgroups of type N_{4} only occur when q is an odd prime. In that case D ∩ N has size exactly 3.2^{3} = 24; see [9]. The +-reflections of rN_{0} are those with centers from an orthonormal basis for V, eigenvectors for Z(S). The corresponding subloop is an elementary abelian group of order 8.

( 4.8) Proposition. If N_{0} has type P_{2} or R_{s2}, then |D ∩ N| is, respectively, 3q^{3}(q − 1)/d or 3q^{3}(q^{2} − 1)/d and so divides |D|.

Proof. Let N_{0} have type R_{s2}. There is a totally singular 2-space T of V with N_{0} its stabilizer in M and N its stabilizer in G. (See [14, p.194].) As T is totally singular, the +-reflection a fixes T if and only if its center Z = [V, a] is perpendicular to T. The subspace T^{⊥} is equal to T ⊕ Y, where Y is a nondegenerate 4-space of +-type. The subspace Y contains q(q^{2} − 1)/d nonsingular 1-spaces of +-type, so T^{⊥} contains q^{2} · q(q^{2} − 1)/d such 1-spaces. Therefore |D ∩ N| = 3q^{3}(q^{2} − 1)/d, a divisor of |D| = 3ℓ(q) = 3q^{3}(q^{4} − 1)/d, as desired.

Suppose N_{0} has type P_{2}. Thus N_{0} is the stabilizer of a singular 1-space U and two totally singular 4-spaces, W_{1} and W_{2}, with U contained in the subspace W_{1} ∩ W_{2} of dimension 3. (See [14, p.192].) In the action of G = P Ω_{8}^{+}(q):Sym(3) on its D_{4} geometry, the subgroup N is the global stabilizer of the set {U, W_{1}, W_{2}}, permuting these as Sym(3).

Consider a +-reflection a of rN_{0}. The element a fixes U and switches W_{1} and W_{2}. The center Z = [V, a] is nonsingular of +-type. The reflection a fixes the 5-space W = W_{1} + W_{2}. As W ≥ W^{⊥} = W_{1} ∩ W_{2}, we must have Z ≤ W. Conversely, for any +-reflection b with [V, b] ≤ W, we have b switching W_{1} and W_{2}, the only two maximal totally singular subspaces of W. Also b fixes U, since U is in b-invariant W_{1} ∩ W_{2}, a totally singular subspace of [V, b]^{⊥} = C_{V} (b). Therefore b ∈ rN_{0}.

Thus |D ∩ rN_{0}| equals the number of 1-spaces in W of +-type. Here W is the perpendicular direct sum of the totally singular 3-space W_{1} ∩ W_{2} and a nondegenerate 2-space X of +-type. The subspace X contains (q − 1)/d nonsingular 1-spaces of +-type, so W contains q^{3} · (q − 1)/d such 1-spaces.

Therefore |D ∩ N| = 3q^{3}(q − 1)/d also divides 3ℓ(q), as desired.

Remark. Corresponding loops of cardinality q^{3}(q^{e} − 1)/d, with e = 1, 2, do exist [9]. In fact, a loop for type P_{2} is a subloop of a loop for type R_{s2}.

( 4.9) Proposition. If N_{0} has type I_{+2}, I_{−2}, or I_{+4}, then |D ∩ N| is, respectively, 12(q − 1)/d, 12(q + 1)/d, or 6q(q^{2} − 1)/d and so divides |D|.

Proof. For type I_{ϵe} with ϵ = ± and e = 2, 4, the group N_{0} is maximal in P O_{8}^{+}(q) (for appropriate q) and leaves invariant a unique decomposition of V as the perpendicular direct sum of 8/e component e-spaces, each nondegenerate of ϵ-type. (See [14, p.177 and 194].)

The reflections of rN_{0} must respect this decomposition. An element that switches two of the components has commutator dimension at least 2 and so is not a reflection. On the other hand, any +-reflection whose center belongs to one of the components leaves the decomposition invariant and so belongs to rN_{0}. For e = 2, this gives |D ∩ rN_{0}| = 4 · (q − ϵ1)/d; and, for e = 4 and ϵ = +, this gives |D ∩ rN_{0}| = 2 · q(q^{2} − 1)/d. In all cases, we have |D ∩ N| a divisor of 3ℓ(q) = 3q^{3}(q^{2} − 1)(q^{2} + 1)/d, as desired.

Remark. The loops corresponding to the cases I_{ϵ2} are in fact subloops of loops coming from the case I_{+4}; see [9].

Theorem 4.1 now follows from Theorem 3.4 and Propositions 4.3-4.9.

Remark. As the remarks of this section indicate, the present arguments come close to, but fall short of, a classification of the maximal subloops of the finite Paige loops. We have given the order of the reflection class in each case. This determines the order of the corresponding subloops, but there could be isotopic, nonisomorphic loops (necessarily of the same order) corresponding to the same class. (In fact, this does not happen.) It is also the case that the various loops may not be maximal. Indeed, the only maximal subloops come under types S_{a}, S_{2}, K_{5}, R_{s2}, and I_{+4}. (When q = 2, a subgroup of type I_{+4} is contained properly in one of type I_{−2}, but the two subgroups correspond to the same maximal subloop of order 12.) See [9, 12, 17] for full discussion and resolution of these questions.

Acknowledgements. We thank Michael Kinyon and Gábor Nagy for bringing the results of [11] to our attention, and we thank Doctors Grishkov and Zavarnitsine for their understanding and encouragement.

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