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On automorphism groups of a biplane (121,16,2)

Dean Crnković, Doris Dumičić Danilović, Sanja Rukavina

Abstract

The existence of a biplane with parameters $(121,16,2)$ is an open problem. Recently, it has been proved by Alavi, Daneshkhah and Praeger that the order of an automorphism group of a of possible biplane ${\mathcal D}$ of order $14$ divides $2^7\cdot3^2\cdot5\cdot7\cdot11\cdot13$. In this paper we show that such a biplane do not have an automorphism of order $11$ or $13$, and thereby establish that $|Aut({\mathcal D})|$ divides $2^7\cdot3^2\cdot5\cdot7.$ Further, we study a possible action of an automorphism of order five or seven, and some small groups of order divisible by five or seven, on a biplane with parameters $(121,16,2)$.

On automorphism groups of a biplane (121,16,2)

Abstract

The existence of a biplane with parameters is an open problem. Recently, it has been proved by Alavi, Daneshkhah and Praeger that the order of an automorphism group of a of possible biplane of order divides . In this paper we show that such a biplane do not have an automorphism of order or , and thereby establish that divides Further, we study a possible action of an automorphism of order five or seven, and some small groups of order divisible by five or seven, on a biplane with parameters .

Paper Structure

This paper contains 8 sections, 7 theorems, 3 equations, 10 tables.

Table of Contents

  1. Introduction
  2. Orbit matrices and automorphisms of order 11 or 13
  3. Orbit matrices for an automorphism of order 13
  4. Orbit matrices for an automorphism of order 11
  5. Automorphisms of order 5 or 7
  6. An action of an automorphism of order $7$
  7. An action of an automorphism of order $5$
  8. Small groups of composite order divisible by 5 or 7

Key Result

Theorem 1.3

Let ${\mathcal{D}}$ be a biplane with parameters $(121,16,2)$. Then $|Aut({\mathcal{D}})|$ divides $2^7\cdot3^2\cdot5\cdot7$. Moreover, if $G\leq Aut({\mathcal{D}})$, then

Theorems & Definitions (9)

  • Theorem 1.3: Main theorem
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 4.1
  • Lemma 4.2
  • Lemma 4.4
  • Lemma 4.5