Chevalley Groups of Type G2 as Automorphism Groups of Loops

Let M∗(q) be the unique nonassociative finite simple Moufang loop constructed over GF (q). We prove that Aut(M∗(2)) is the Chevalley group G2(2), by extending multiplicative automorphism of M∗(2) into linear automorphisms of the unique split octonion algebra over GF (2). Many of our auxiliary results apply in the general case. In the course of the proof we show that every element of a split octonion algebra can be written as a sum of two elements of norm one. 1 Composition Algebras and Paige Loops Let C be a finite-dimensional vector space over a field k, equipped with a quadratic form N : C −→ k and a multiplicative operation ·. Following [7], we say that C = (C, N, +, ·) is a composition algebra if (C, +, ·) is a nonassociative ring with identity element e, N is nondegenerate, and N(u · v) = N(u)N(v) is satisfied for every u, v ∈ C. The bilinear form associated with N will also be denoted by N . Recall /that N : C × C −→ k is defined by N(u, v) = N(u + v) − N(u) − N(v). Write u ⊥ v if N(u, v) = 0, and set u⊥ = {v ∈ C; u ⊥ v}. The standard 8-dimensional real Cayley algebra O constructed by the CayleyDickson process (or doubling [7]) is the best known nonassociative composition algebra. There is a remarkably compact way of constructing O that avoids the iterative Cayley-Dickson process. As in [2], let B = {e = e0, e1, . . ., e7} be a basis whose vectors are multiplied according to er = −1, er+7 = er, eres = −eser, er+1er+3 = er+2er+6 = er+4er+5 = er, (1) for r, s ∈ {1, . . . , 7}, r 6= s. (Alternatively, see [1, p. 122].) The norm N(u) of a vector u = ∑7 i=0 aiei ∈ O is given by ∑7 i=0 a 2 i . Importantly, all the structural constants γijk, defined by ei · ej = ∑7 k=0 γijkek, are equal to ±1, and therefore the construction can be imitated over any field k. For k = GF (q) of odd characteristic, let us denote the ensuing algebra by O(q). When q is even, the above construction does not yield a composition algebra. The following facts about composition algebras can be found in [7]. Every nontrivial composition algebra C has dimension 2, 4 or 8, and we speak of a complex, quaternion or octonion algebra, respectively. We say that C is a division algebra if it has no zero divisors, else C is called split. There can be many non-isomorphic octonion algebras over a given field. Exactly one of them is guaranteed to be split. Moreover, when k is finite, all octonion algebras over k are isomorphic (and thus split). Let O(q) be the unique octonion algebra constructed over GF (q). Automorphism Groups of Paige Loops 2 All composition algebras satisfy the so-called Moufang identities (xy)(zx) = x((yz)x), x(y(xz)) = ((xy)x)z, x(y(zy)) = ((xy)z)y. (2) These identities are the essence of Moufang loops, undoubtedly the most investigated variety of nonassociative loops. More precisely, a quasigroup (L, ·) is a Moufang loop if it possesses a neutral element e and satisfies one (and hence all) of the Moufang identities (2). We refer the reader to [6] for the basic properties of loops and Moufang loops in particular. Briefly, every element x of a Moufang loop L has a both-sided inverse x−1, and a subloop 〈x, y, z〉 of L generated by x, y and z is a group if and only if x, y and z associate. Specifically, every two-generated subloop of L is a group. Paige [5] constructed one nonassociative finite simple Moufang loop for every finite field GF (q). Liebeck [3] used the classification of finite simple groups in order to prove that there are no other nonassociative finite simple Moufang loops. Reflecting the current trend in loop theory, we will call these loops Paige loops, and we denote the unique Paige loop constructed over GF (q) by M∗(q). The relation between O(q) and M∗(q) is as follows. Let M(q) be the set of all elements ofO(q) of norm one. Then M(q) is a Moufang loop with center Z(M(q)) = {e, −e}, and M(q)/Z(M(q)) is isomorphic to M∗(q). Note that M(q) = M∗(q) in characteristic 2. Historically, all split octonion algebras and Paige loops were constructed by Zorn [9] and Paige without reference to doubling. Given a field k, consider the vector matrix algebra consisting of all vector matrices