### 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 language | English (US) |
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Pages (from-to) | 618-652 |

Number of pages | 35 |

Journal | Advances in Mathematics |

Volume | 234 |

DOIs | |

State | Published - 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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## Cite this

*Advances in Mathematics*,

*234*, 618-652. https://doi.org/10.1016/j.aim.2012.11.005