TY - JOUR

T1 - Tactical decompositions and orbits of projective groups

AU - Cameron, P. J.

AU - Liebler, R. A.

PY - 1982/1/1

Y1 - 1982/1/1

N2 - We consider decompositions of the incidence structure of points and lines of PG(n, q) (n≥3) with equally many point and line classes. Such a decomposition, if line-tactical, must also be point-tactical. (This holds more generally in any 2-design.) We conjecture that such a tactical decomposition with more than one class has either a singleton point class, or just two point classes, one of which is a hyperplane. Using the previously mentioned result, we reduce the conjecture to the case n=3, and prove it when q2+q+1 is prime and for very small values of q. The truth of the conjecture would imply that an irreducible collineation group of PG(n, q) (n≥3) with equally many point and line orbits is line-transitive (and hence known).

AB - We consider decompositions of the incidence structure of points and lines of PG(n, q) (n≥3) with equally many point and line classes. Such a decomposition, if line-tactical, must also be point-tactical. (This holds more generally in any 2-design.) We conjecture that such a tactical decomposition with more than one class has either a singleton point class, or just two point classes, one of which is a hyperplane. Using the previously mentioned result, we reduce the conjecture to the case n=3, and prove it when q2+q+1 is prime and for very small values of q. The truth of the conjecture would imply that an irreducible collineation group of PG(n, q) (n≥3) with equally many point and line orbits is line-transitive (and hence known).

UR - http://www.scopus.com/inward/record.url?scp=0008980446&partnerID=8YFLogxK

U2 - 10.1016/0024-3795(82)90029-5

DO - 10.1016/0024-3795(82)90029-5

M3 - Article

AN - SCOPUS:0008980446

SN - 0024-3795

VL - 46

SP - 91

EP - 102

JO - Linear Algebra and its Applications

JF - Linear Algebra and its Applications

IS - C

ER -