A note on the conjecture of the autotopism group of the Figueroa’s presemifields

Authors

  • Walter J. Meléndez Florián Facultad de Ciencias Físicas y Matemáticas, Universidad Nacional de Trujillo, Avenida Juan Pablo II s/n, Trujillo, Perú. http://orcid.org/0000-0002-6253-0205
  • Moisés R. Delgado Olórtegui Mathematics and Physics Department, University of Puerto Rico, Cayey Campus, 205 Calle Antonio R. Barceló, Cayey, Puerto Rico. http://orcid.org/0000-0002-5267-8169

DOI:

https://doi.org/10.17268/sel.mat.2020.02.09

Keywords:

Finite presemifield, finite semifield, autotopism group, Cordero-Figueroa semifield, Figueroa’s presemifield

Abstract

In [4] was stated the following conjecture: If a Figueroa’s presemifield P(K; alfa; beta; A;B) admits an autotopism of order a p-primitive prime divisor of p^n-1, then its autotopism group is isomorphic to a subgroup of GL(K) x GL(K). In [5] this conjecture was settled under an additional normality condition. In this article, we show that the assumption in the hypothesis of the conjecture is necessary in the sense that there exist a Figueroa’s presemifield, that does not admit such autotopism, for which the conjecture is not met.

Author Biographies

Walter J. Meléndez Florián, Facultad de Ciencias Físicas y Matemáticas, Universidad Nacional de Trujillo, Avenida Juan Pablo II s/n, Trujillo, Perú.

Departamento de Matemáticas

Profesor Auxiliar

Moisés R. Delgado Olórtegui, Mathematics and Physics Department, University of Puerto Rico, Cayey Campus, 205 Calle Antonio R. Barceló, Cayey, Puerto Rico.

Mathematics and Physics Deparment

Assistant Professor

References

Cordero M, Figueroa R. Towards a characterization of the generalized twisted field planes. J. Geom. 1995; 52(1-2):54-63.

Dempwolff U. Semifields Planes of Order 81. J. Geom. 2008; 89(1-2):1-16.

Hughes D, Piper F. Projective Planes. New York: Springer-Verlag; 1973.

Meléndez W, Figueroa R, Delgado M. On the autotopism group of the Cordero-Figueroa semifield of order 36. Discuss. Math.-General Algebra and Appl. 2016; 36(1):117-126. doi:10.7151/dmgaa.1250.

Meléndez W, Delgado M. On a conjecture about the autotopism group of the Figueroa’s presemifield of order pn. Note Mat. 2018; 38(2):11-20. doi: 10.1285/i15900932v38n2p11.

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Published

2020-12-25

How to Cite

Meléndez Florián, W. J., & Delgado Olórtegui, M. R. (2020). A note on the conjecture of the autotopism group of the Figueroa’s presemifields. Selecciones Matemáticas, 7(02), 267-275. https://doi.org/10.17268/sel.mat.2020.02.09

Issue

Vol. 7 No. 02 (2020): August - December

Section

Articles

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