zbMATH — the first resource for mathematics

Semifield planes of odd order that admit a subgroup of autotopisms isomorphic to $$A_4$$. (English. Russian original) Zbl 1358.51005
Russ. Math. 60, No. 9, 7-22 (2016); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2016, No. 9, 10-25 (2016).
The author describes odd-order semifield planes admitting a Baer involution or an alternating group $$A_{4}$$ in the translation complement.
Explicit matrix representations of the spreads are given. The results are applied to the order $$81$$ case.
The reviewer is surprised not to find any references to papers after the year 2000, from outside the authors work group. Certainly relevant would be, e.g., [U. Dempwolff, J. Geom. 89, No. 1–2, 1–16 (2008; Zbl 1175.12003)].

MSC:
 51E15 Finite affine and projective planes (geometric aspects) 51A40 Translation planes and spreads in linear incidence geometry 51E23 Spreads and packing problems in finite geometry
Full Text:
References:
 [1] Biliotti, M.; Jha, V.; Johnson, N. L.; Menichetti, G., A structure theory for two-dimensional translation planes of order q2 that admit collineation group of order q2, Geom.Dedic., 29, 7-43, (1989) · Zbl 0673.51003 [2] Huang, H.; Johnson, N. L., 8 semifield planes of order 82, DiscreteMath., 80, 69-79, (1990) · Zbl 0699.51003 [3] Cordero, M., Matrix spread sets of p-primitive semifield planes, Int. J. Math. and Math. Sci., 20, 293-298, (1997) · Zbl 0885.51008 [4] Podufalov, N. D.; Durakov, B. K.; Kravtsova, O. V.; Durakov, E. B., On semifield planes of order 162, Sib. Math. J., 37, 535-541, (1996) · Zbl 0879.51004 [5] Kravtsova, O. V., Some subgroups of automorphisms of semifield planes, 78-80, (2013) [6] Kravtsova, O. V., A subgroup of autotopisms of anoddorder semifield plane isomorphic to the alternating group $$A$$_{4}, 89, (2014) [7] Hughes, D. R. and Piper, F. C. Projective Planes (Springer-Verlag, New York, 1973). · Zbl 0267.50018 [8] Podufalov, N. D., On spread sets and collineations of projective planes, Contem.Math., 131, 697-705, (1992) · Zbl 0779.51001 [9] Unsolved Problems of the Theory of Groups. Kourov Notebook. 16-th Edition, complemented and including the archive of solved problems. Ed. by V. D. Mazurov and E. I. Khukhro (Novosibirsk, 2006). · Zbl 1084.20001 [10] Levchuk, V. M.; Panov, S. V.; Stukkert, P. K., Questions of classification of projective planes and Latin rectangles, 56-70, (2012) [11] Kravtsova, O. V., Semifield planes of even order that admit the Baer involution, Izv. Irkutsk.Gos. Univ., Ser. Mat., 6, 26-37, (2013) · Zbl 1295.51006 [12] Kravtsova, O. V.; Panov, S. V.; Shevelyova, I. V., Some results on isomorphisms of finite semifield planes, J. Siberian Federal University.Mathematics & Physics, 6, 33-39, (2013) [13] Kravtsova, O. V.; Kurshakova, P. K., On the question of isomorphity of semifield planes, Vestnik KGTU.Matem.Metody iModelir., 42, 13-19, (2006) [14] Podufalov, N. D.; Busarkina, I. V.; Durakov, B. K., On the autotopism group of a semifield p-primitive plane, 190-195, (1998) [15] Kravtsova, O. V., On some translation planes admitting $$A$$_{4}, 38-39, (2004) [16] Kravtsova, O. V.; Pramzina, V. O., On a subgroup of collineations of a semifield plane isomorphic to $$A$$_{4}, J. Siberian Federal University. Mathematics & Physics, 4, 498-504, (2011)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.