The groups A7, A8 and a projective plane of order 16

Three results about the groups A7 , As) L3(2), L4(2) and the Lorimer-Rahilly plane are proved in a unified way. My principal contact with Alan Rahilly was through a certain projective plane of order 16. He had found it as one of an infinite class of generalized Hall planes he constructed in his thesis at Sydney University [9], while I had found it as an isolated plane which I identified through the isomorphism between the groups As and L4(2). [5]. It is now called the Lorimer-Rahilly plane. It has played a critical part in the classification of projective planes in that it has many interesting properties : for example, it is the only known plane of type (6,m), as described by D.R. Hughes; it has type (6,2) and the only other value of m possible is 3 (6]. While Alan constructed an infinite class of planes, the one I constructed appears to be the most interesting of them. My original proof depended on a certain isomorphism between the groups As and L4(2), but I gave a more direct one in [7]. However, much more turns out to be possible. By straight-forward elementary arguments it is possible, in a unified manner, to 1. prove that As and L4(2) are isomorphic, 2. prove that A7 has a 2-transitive representation of degree 15, 3. construct the Lorimer-Rahilly plane and present a large part of its collineation group. The discovery that the groups As and L4 (2) are isomorphic seems to have been made by Jordan and it appears in his classic, "Traite des Substitutions et les Equations Algebriques" [4], where the proof is given in terms of the Galois group of a polynomial equation of degree 8. A group-theoretic proof was given by Moore [8], Australasian Journal of Combinatorics ~(1993), pp .45-51 who showed, in the same paper, the fact about A7 which appears here as Proposition 6. It was taken up again, by Dickson, and his proof appears in [IJ. While none of these results is new, it seems appropriate to dedicate their unification to the memory of Alan Rahilly. 1. The groups L3(2), A7 , L4(2) and Aso The purpose of this section is to show that the groups As and L4(2) are isomorphic and that A7 , as a subgroup of L4(2), has a 2-transitive representation of degree 15. The story begins with the eight numbers 0,1,2, ... , 7, taken as the vectors of a 3-dimensional vector space, V3 , over the field of order 2. To be definite, let us take the following as the sets of non-zero vectors of the 2-dimensional subspaces of V3 {I, 2, 3}, {3, 4, 5}, {5, 6, I}, {I, 7, 4}, {3, 7, 6}, {5, 7, 2}, {2, 4, 6}. Otherwise described, these are the lines of a projective plane of order 2, which will be denoted by 7l"z. In the sequel, As will be taken as the alternating group on the vectors of V3 , A7 will be the stabilizer of 0 in As, L3 (2) will be the full linear group of V3 and T will be the group of translations of V3 . Thus, L3 (2) is a subgroup of A7 , T is a complement of A7 in As and TL3(2) is a subgroup of As of index 15. The key to this paper is a set, M, of permutations of As defined from the 2dimensional su bspaces of V3 (or from the lines of 7l"z): M {I, (123), (132), (345), (354), (561), (516), (174), (147), (376), (367), (572), (527), (246), (264)}. The principal property of M is described in the first Proposition. Proposition 1. M is a set of representatives of the left cosets ofT L3(2) in As and of L3(2) in A7 • Proof. If ml, ffiZ are two different members of M, then mi""lmz is a 3-cycle or a 5-cycle. On the other hand, T L3(2) contains no such element. Thus, the members of M lie in different cosets of T L3 (2) and, as this subgroup has index 15 in As, they form a set of coset representatives. As M ~ A7 ) they also form a set of coset representatives of L3(2) in A7 . That proves Proposition 1. Proposition 1 defines a representation of the group As as a permutation group on M: if 9 E A 8 , define g: M -+ M by the equation g(m)TL3(2) = gmTL3(2). The next Proposition calculates this representation explicitly for some particular members of A 8 •