Equally partitioned groups and projective planes of order 9 (Q1078323)

The authors show that the elementary abelian group of order 81 has exactly 8424 partitions in subgroups of order 9; under the group GL(4,3) these partitions fall in two equivalence classes of length 2106 and 6318. This gives another proof of the well-known result that there are exactly two translation planes of order 9 [cf. \textit{M. Hall}, Trans. Am. Math. Soc. 54, 229-277 (1943; Zbl 0060.32209), Appendix II; or \textit{H. Lüneburg}, Translation Planes (1980; Zbl 0446.51003), 8.4, p. 36].