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Unisingular subgroups of symplectic group $Sp_{2n}(2)$ for $2n<250$

Alexandre Zalesski

Abstract

A linear group is called unisingular if every element of it has eigenvalue 1. A certain aspect of the theory of abelian varieties requires the knowledge of unisingular irreducible subgroups of the symplectic groups over the field of two elements. A more special, but an important question is on the existence of such subgroups in the symplectic groups of particular degree. We answer this question for almost all degrees $2n<250$, specifically, the question remains open only 7 values of $n$. Additionally, the paper contains results of general nature on the structure of unisingular irreducible linear groups.

Unisingular subgroups of symplectic group $Sp_{2n}(2)$ for $2n<250$

Abstract

A linear group is called unisingular if every element of it has eigenvalue 1. A certain aspect of the theory of abelian varieties requires the knowledge of unisingular irreducible subgroups of the symplectic groups over the field of two elements. A more special, but an important question is on the existence of such subgroups in the symplectic groups of particular degree. We answer this question for almost all degrees , specifically, the question remains open only 7 values of . Additionally, the paper contains results of general nature on the structure of unisingular irreducible linear groups.
Paper Structure (20 sections, 80 theorems, 9 equations, 1 table)

This paper contains 20 sections, 80 theorems, 9 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Some general results
  4. Remarks on Clifford's theory
  5. Unisingular groups and subspace covering problems
  6. Unisingular representations of simple groups $L_2(q)$
  7. Observations on representations of some groups of Lie type
  8. Some simple irreducible linear groups
  9. The multiplicity of $1_G$ in the 2-modular reduction of the Weil representations of $SU_n(q)$, $q$ even
  10. The Steinberg characters of unitary groups $SU_{n}(q), n$ odd, $q$ even, and $E_7(q)$, $q$ even
  11. Examples of irreducible unisingular subgroups
  12. Sources of examples
  13. The cases $2n=12,30$
  14. The cases with $2n\in\{22,46, 82,146\}$
  15. Cases with $2n\in\{174,198,202,218, 246 \}$
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $G$ be a finite simple group of Lie type in defining characteristic $2$. Suppose that $G$ is not isomorphic to $PSL_2(q)$ for $q$ even. Then $G$ is isomorphic to a unisingular irreducible subgroup of $Sp_{2n}(2)$ with $2n=|G|_2$, where $|G|_2$ is the $2$-part of $|G|$.

Theorems & Definitions (148)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 1.3
  • Lemma 1.4
  • Lemma 1.5
  • Theorem 1.6
  • Lemma 1.7
  • Theorem 1.8
  • Lemma 2.1
  • Lemma 2.2
  • ...and 138 more