A property of A7, and a maximal 3-dimensional linear section of GL(4, 2) Original Research Article

A 7-cluster in A7 is defined to be a 7-subset II = { ξ1, ξ2, … ξ7} of A7 such that ξ1−1 ξ, is of order 6 for all i ≠ j; a normalized 7-cluster is one which contains the identity. Under the action by conjugation of A7, normalized 7-clusters fall into two orbits O, O′, each of length 105, and a simple recipe is given for their construction. A linear section of GL(n,q) is a linear subspace S of End(n,q) such that every nonzero element of S lies in GL(n, q). It is shown, for normalized clusters H belonging to one, but not both, of the orbits O. O′, that if T maps A8 isomorphically onto GL(4, 2) then the 7 elements of T(Π) are the nonzero elements of a maximal 3-dimensional linear section of GL(4,2). © 1999 Elsevier Science B.V. All rights reserved