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Character covering number of $\mathrm{PSL}_2 (q)$

Namrata Arvind, Saikat Panja

Abstract

For a group $G$ and a character $χ$ of $G$, let $c(χ)$ denote the set of all irreducible characters of $G$, occurring in $χ$. We prove that whenever $q\geq 8$, all non-trivial irreducible character $χ$ of $\mathrm{PSL}_2(q)$ satisfies $c(χ^4)=\mathrm{Irr}\left(\mathrm{PSL}_2(q)\right)$ if $q=2^{2m+1}$ and $c(χ^3)=\mathrm{Irr}\left(\mathrm{PSL}_2(q)\right)$ otherwise.

Character covering number of $\mathrm{PSL}_2 (q)$

Abstract

For a group and a character of , let denote the set of all irreducible characters of , occurring in . We prove that whenever , all non-trivial irreducible character of satisfies if and otherwise.
Paper Structure (8 sections, 4 theorems, 32 equations, 5 tables)

This paper contains 8 sections, 4 theorems, 32 equations, 5 tables.

Table of Contents

  1. Introduction
  2. Some background results
  3. Conjugacy classes and character table
  4. Proof of \ref{['thm:main']}
  5. When $q$ is even
  6. When $q$ is odd
  7. Case I: $q\equiv 3\pmod{4}$
  8. Case II: $q\equiv 1\pmod{4}$

Key Result

Theorem 1.1

Let $q\geq 8$ be a power of prime and $G(q)=\mathrm{PSL}_2(q)$ be the projective special linear group (which is also simple). Then the covering number is $4$ if $q=2^{2m+1}$ for some $m$ and $3$ otherwise.

Theorems & Definitions (7)

  • Theorem 1.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof