LINEAR SECTIONS OF GL(4,2)

For V = V(n, q), a linear section of GL(V) = GL(n, q) is a vector subs pace S of the n(2)-dimensional vector space End(V) which is contained in GL(V) U {0}. We pose the problem, for given (n, q), of classifying the different kinds of maximal linear sections of GL(n, q). If S is an y linear section of GL(n, q) then dim S less than or equal to n. The c ase of GL(4, 2) is examined fully. Up to a suitable notion of equivale nce there are just two classes of S-dimensional maximal normalized lin ear sections M-3, M-3', and three classes M-4, M-4', M-4'' of 4-dimens ional sections. The subgroups of GL(4, 2) generated by representatives of these five classes are respectively G(3) congruent to A(7), G(3)' = GL(4, 2), G(4) congruent to Z(15), G4' congruent to Z(3) x A(5), G(4 )'' = GL(4, 2). On various occasions use is made of an isomorphism T : As --> GL(4, 2). In particular a representative of the class M3 is th e image under T of a subset {xi(1), ... ,xi(7)} of A(7) with the prope rty that xi(i)(-1)xi(j) is of order 6 for all i not equal j. The class es M-3, M-3' give rise to two classes of maximal partial spreads of or der 9 in PG(7, 2), and the classes M-4', M-4'' yield the two isomorphi sm classes of proper semifield planes of order 16.