Linear sections of GL(4, 2)

Gordon, Neil A.; Lunardon, Guglielmo; Shaw, Ron

Abstract

For V = V (n; q); a linear section of GL(V ) = GL(n; q) is a vector subspace S of the n 2 -dimensional vector space End(V ) which is contained in GL(V ) [ f0g: We pose the problem, for given (n; q); of classifying the di erent kinds of maximal linear sections of GL(n; q): If S is any linear section of GL(n; q) then dim S n: The case of GL(4; 2) is examined fully. Up to a suitable notion of equiv- alence there are just two classes of 3-dimensional maximal normalized linear sections M3;M0 3 , and three classes M4;M0 4 ;M00 4 of 4-dimensional sections. The subgroups of GL(4; 2) generated by representatives of these ve classes are respectively G3 = A7; G 0 3 = GL(4; 2); G4 = Z15; G 0 4 = Z3 A5; G 00 4 = GL(4; 2): On various occasions use is made of an isomorphism T : A8 ! GL(4; 2): In particular a representative of the class M3 is the image under T of a subset f1; ::: ; 7g of A7 with the property that 1 i j is of order 6 for all i =6 j: The classes M3;M0 3 give rise to two classes of maximal partial spreads of order 9 in PG(7; 2); and the classes M0 4 ;M00 4 yield the two isomorphism classes of proper semi eld planes of order 16.

• Journal Article Type: Article
• Publication Date: 1998
• Journal: Bulletin of the Belgian Mathematical Society - Simon Stevin
• Print ISSN: 1370-1444
• Peer Reviewed: Peer Reviewed
• Volume: 5
• Issue: 2-3
• Pages: 287-311
• APA6 Citation: Gordon, N. A., Lunardon, G., & Shaw, R. (1998). Linear sections of GL(4, 2). Bulletin of the Belgian Mathematical Society, Simon Stevin, 5(2-3), 287-311. https://doi.org/10.36045/bbms/1103409012
• DOI: https://doi.org/10.36045/bbms/1103409012
• Keywords: Finite geometry; Linear groups; Partial spreads; Semifield planes
• Publisher URL: https://projecteuclid.org/euclid.bbms/1103409012