The Translation Planes of Order Twenty-Five

The purpose of this paper is to classify the translation planes of order twenty-five. Andre [l] and Bruck and Bose [3] have shown there is a one-one correspondence between spreads in projective space and translation planes. We use this fact to classify the spreads in the three-dimensional projective space PG(3, q). The translation planes of order 16 have been classified by Dempwolff and Reifart [9, 191. Every spread in PG(3, q) may be obtained by replacing a subset of the lines of a regular spread. The classification is divided into two parts. In the first part we classify those translation planes of order 25 containing a regulus and in the second part we classify the translation planes that do not contain a regulus. Our results show that there are 21 isomorphism classes of translation planes of order 25; 13 are planes containing a regulus and eight planes not containing a regulus. Of the 21 classes, seven were previously unknown. The results of the first part of this paper form part of a doctoral thesis (Oakden’s) [ 131 completed under the direction of F. A. Sherk in 1973.