On (G,\(\Gamma\) ,n,q)-translation planes (Q760905)

A translation plane \(\pi\) is a (G,\(\Gamma\),n,q)-plane if it is defined by a spread \(\Gamma\) on a vector space V of dimension 2n over GF(q) and admits a group G in the translation complement such that G leaves a set \(\Delta\) of \((q+1)\) components of \(\Gamma\) invariant and is transitive on \(\Gamma\)-\(\Delta\). There are several known classes of examples. The main result of this paper is the following: Let \(\pi\) be a (G,\(\Gamma\),n,q)-plane of characteristic p. Then one of the following conditions holds: (1) \(0_ p(G)\) is semi-regular on \(\Delta\)-\(\{\) \(A\}\) for some \(A\in \Delta\). (2) \(n=2\). (3) \(n=3\) and q is odd. Other conditions are also developed. For instance if \(q^ 2\equiv -1\) mod 4 and \(0_ p(G)\neq 1\), then \(n=3\).