Linear spaces with significant characteristic prime
نویسنده
چکیده
Let G be a group with socle a simple group of Lie type defined over the finite field with q elements where q is a power of the prime p. Suppose that G acts transitively upon the lines of a linear space S. We show that if p is significant then G acts flag-transitively on S and all examples are known. MSC(2000): 20B25, 05B05. 1 Background and statement of result A linear space S is an incidence structure of points and lines such that any two points are incident with exactly one line. Also S is non-trivial provided any point is incident with at least two lines and any line is incident with at least two points; all linear spaces considered in this paper will be presumed to be non-trivial. A flag is a pair (α, L) where α is a point incident with a line L. Let S be a finite linear space admitting an automorphism group G which is transitive on lines. Then S is said to have parameters b (the number of lines), v (the number of points), k (the number of points incident with a line) and r (the number of lines incident with a point). Camina, Neumann and Praeger [CNP03] have defined a prime p to be significant for the space S if it divides into (b, v − 1). They then show that if P is a Sylow p-subgroup of G and Gα is a point-stabilizer in G then Gα ≥ NG(P ) [CNP03, Lemma 6.1]. The finite linear spaces which admit a flag-transitive almost simple group have been classified in [Kle90, Sax02]. As part of the program to extend this classification to those linear spaces which admit a line-transitive almost simple group we prove the following theorem:
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