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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
MSC:
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
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References:
[1] I. M. Isaacs, ”Equally partitioned groups,” Pac. J. Math.,49, No. 1, 109–116 (1973). · Zbl 0235.20021
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