Line-transitive point-imprimitive linear spaces with number of points being a product of two primes

The work studies the line-transitive point-imprimitive automorphism groups of finite linear spaces, and is underway on the situation when the numbers of points are products of two primes. Let [Formula: see text] be a non-trivial finite linear space with [Formula: see text] points, where [Formula: see text] and [Formula: see text] are two primes. We prove that if [Formula: see text] is line-transitive point-imprimitive, then [Formula: see text] is solvable.

A dual approach to embedding the complement of two lines in a finite projective plane

Let S be a finite linear space on v ≥ n2 –n points and b = n2+n+1–m lines, m ≧ 0, n ≧ 1, such that at most m points are not on n + 1 lines. If m ≧ 1, except if m = 1 and a unique point on n lines is on no line with two points, then S embeds uniquely in a projective plane of order n or is one exceptional case if n =4. If m ≦ 1 and if v ≧ n2 – 2√n + 3, + 6, the same conclusion holds, except possibly for the uniqueness.1991 Mathematics subject classification (Amer. Math. Soc.) 05 B 05, 51 E 10.

The exceptional case in a theorem of Bose and Shrikhande

It is shown that (n being an integer) any non-trivial finite linear space with n2- 1 points, all of degree at most n+1, is embeddable in a finite projective plane of order n. This generalizes a theorem of Bose and Shrikhande and settles the unsolved case n = 6.

Embedding the complement of two lines in a finite projective plane

In this paper we use a result from graph theory on the characterization of the line graphs of the complete bigraphs to show that if n is any integer ≥ 2 then any finite linear space having p = n2 − n or p = n2 − n + 1 points, of which at least n2 − n have degree n + 1, and q ≤ n2 + n − 1 lines is embeddable in an FPP of order n unless n = 4. If n = 4 there is only one possible exception for each of the two values of p, and for p = n2 − n, this exception can be embedded in the FPP of order 5.