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 -