ABSTRACT
R em ark 6 .1 Every PG(2,q) admits a cyclic subgroup G (called a 1Singer cycle’) of order q2 +q+l acting regularly on the points (lines) of the projective plane.
Proof: Let GF(q3) be a 3-dimensional vector space over GF(q) and note that there exists a cyclic group G of order q3 — 1 of GF(q3) — {0}. Hence, G acts on PG (2 , q), is transitive and hence regular or order (q3 — 1 )/(q — 1) on the set of 1-dimensional GF(g)-subspaces of GF(q3) which forms the points of PG(2,g).D
Lem m a 6.2 Let K be a right nearfield of order q2 which contains a field F ~ GF(q) in its center.
