Products of conjugacy classes in finite and algebraic simple groups

Robert M. Guralnick, Gunter Malle, Pham Huu Tiep

Research output: Contribution to journalArticle

11 Scopus citations

Abstract

We prove the Arad-Herzog conjecture for various families of finite simple groups - if A and B are nontrivial conjugacy classes, then AB is not a conjugacy class. We also prove that if G is a finite simple group of Lie type and A and B are nontrivial conjugacy classes, either both semisimple or both unipotent, then AB is not a conjugacy class. We also prove a strong version of the Arad-Herzog conjecture for simple algebraic groups and in particular show that almost always the product of two conjugacy classes in a simple algebraic group consists of infinitely many conjugacy classes. As a consequence we obtain a complete classification of pairs of centralizers in a simple algebraic group which have dense product. A special case of this has been used by Prasad to prove a uniqueness result for Tits systems in quasi-reductive groups. Our final result is a generalization of the Baer-Suzuki theorem for p-elements with p'5.

Original languageEnglish (US)
Pages (from-to)618-652
Number of pages35
JournalAdvances in Mathematics
Volume234
DOIs
StatePublished - Feb 5 2013

Keywords

  • Algebraic groups
  • Baer-Suzuki theorem
  • Characters
  • Finite simple groups
  • Products of centralizers
  • Products of conjugacy classes
  • Szep's conjecture

ASJC Scopus subject areas

  • Mathematics(all)

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