### 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) |
---|---|

Pages (from-to) | 618-652 |

Number of pages | 35 |

Journal | Advances in Mathematics |

Volume | 234 |

DOIs | |

State | Published - Feb 5 2013 |

### Fingerprint

### 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)

### Cite this

*Advances in Mathematics*,

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

**Products of conjugacy classes in finite and algebraic simple groups.** / Guralnick, Robert M.; Malle, Gunter; Tiep, Pham Huu.

Research output: Contribution to journal › Article

*Advances in Mathematics*, vol. 234, pp. 618-652. https://doi.org/10.1016/j.aim.2012.11.005

}

TY - JOUR

T1 - Products of conjugacy classes in finite and algebraic simple groups

AU - Guralnick, Robert M.

AU - Malle, Gunter

AU - Tiep, Pham Huu

PY - 2013/2/5

Y1 - 2013/2/5

N2 - 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.

AB - 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.

KW - Algebraic groups

KW - Baer-Suzuki theorem

KW - Characters

KW - Finite simple groups

KW - Products of centralizers

KW - Products of conjugacy classes

KW - Szep's conjecture

UR - http://www.scopus.com/inward/record.url?scp=84870791056&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84870791056&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2012.11.005

DO - 10.1016/j.aim.2012.11.005

M3 - Article

VL - 234

SP - 618

EP - 652

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

ER -