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The autotopism group of a family of commutative semifields

Lukas Kölsch, Alexandra Levinshteyn, Milan Tenn

TL;DR

This work determines the autotopism group of a prominent large family of commutative finite semifields introduced by Göloğlu and Kölsch, showing that all autotopisms are semilinear over a degree-2 subfield and that the group is solvable. The authors reduce the problem to monomial subfunctions via nuclei-based semilinear constraints and then perform a detailed diagonal/monomial analysis to obtain a complete classification: Aut(C(S)) is isomorphic to $\mathbb{Z}_{p^m-1}\times\mathbb{Z}_{p^e-1}$ in general, with an additional $\mathbb{Z}_2$ factor and a semidirect component $\rtimes \mathbb{Z}_{m/i}$ when $A^2$ is an $(r-1)$-st power. The automorphism groups of the associated rank-metric codes and the collineation groups of the corresponding translation planes are determined accordingly. This result provides a strong, explicit handle on isotopism questions for these semifields and contributes evidence toward the conjecture that autotopism groups of finite semifields are solvable.

Abstract

We completely determine the autotopism group of the (as of now) largest family of commutative semifields found by Göloğlu and Kölsch. Since this family of semifields generally does not have large nuclei, this process is considerably harder than for families considered in preceding work. Our results show that all autotopisms are semilinear over the degree 2 subfield and that the autotopism group is always solvable. Using known connections, our results also completely determine the automorphism groups of the associated rank-metric codes and the collineation groups of the associated translation planes.

The autotopism group of a family of commutative semifields

TL;DR

This work determines the autotopism group of a prominent large family of commutative finite semifields introduced by Göloğlu and Kölsch, showing that all autotopisms are semilinear over a degree-2 subfield and that the group is solvable. The authors reduce the problem to monomial subfunctions via nuclei-based semilinear constraints and then perform a detailed diagonal/monomial analysis to obtain a complete classification: Aut(C(S)) is isomorphic to in general, with an additional factor and a semidirect component when is an -st power. The automorphism groups of the associated rank-metric codes and the collineation groups of the corresponding translation planes are determined accordingly. This result provides a strong, explicit handle on isotopism questions for these semifields and contributes evidence toward the conjecture that autotopism groups of finite semifields are solvable.

Abstract

We completely determine the autotopism group of the (as of now) largest family of commutative semifields found by Göloğlu and Kölsch. Since this family of semifields generally does not have large nuclei, this process is considerably harder than for families considered in preceding work. Our results show that all autotopisms are semilinear over the degree 2 subfield and that the autotopism group is always solvable. Using known connections, our results also completely determine the automorphism groups of the associated rank-metric codes and the collineation groups of the associated translation planes.

Paper Structure

This paper contains 7 sections, 21 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. Notation, overview, and preliminaries
  3. The monomial case
  4. The diagonal case
  5. The general case
  6. Putting everything together: The proof of Theorem \ref{['thm:main']}
  7. Conclusion

Key Result

Theorem 1.3

Define a binary operation $\circ$ on $\mathbb F_{p}^{2m}\cong \mathbb F_{p^m}\times \mathbb F_{p^m}$ via where $p$ is odd, $q=p^k$ for some $1 \leq k \leq m-1$, $r=p^{k+m/2}$, $B \in \mathbb F_{p^m}$ is a non-square, $A \in \mathbb F_{p^m}$ such that $AB \in \mathbb F_{p^{m/2}}^\times$, and $m/\gcd(k,m)$ is odd. Then $\mathbb S=(\mathbb F_{p}^{2m}\cong \mathbb F_{p^m}\times \mathbb F_{p^m},+,\cir

Theorems & Definitions (39)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3: golouglu2023exponential
  • Theorem 1.4
  • Theorem 1.5: Main result
  • Lemma 2.2
  • Lemma 2.3
  • Theorem 2.4
  • proof
  • Proposition 3.1
  • ...and 29 more