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Equally partitioned groups and projective planes of order 9. (English. Russian original) Zbl 0596.20020
Algebra Logic 24, 155-163 (1985); translation from Algebra Logika 24, No. 3, 255-266 (1985).
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. M. Hall, Trans. Am. Math. Soc. 54, 229-277 (1943; Zbl 0060.32209), Appendix II; or H. Lüneburg, Translation Planes (1980; Zbl 0446.51003), 8.4, p. 36].
Reviewer: Th.Grundhöfer
20D60 Arithmetic and combinatorial problems involving abstract finite groups
20K01 Finite abelian groups
51A40 Translation planes and spreads in linear incidence geometry
20D30 Series and lattices of subgroups
Full Text: DOI EuDML
[1] I. M. Isaacs, ”Equally partitioned groups,” Pac. J. Math.,49, No. 1, 109–116 (1973). · Zbl 0235.20021
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