### Abstract

Let G be a finite quasisimple group of Lie type. We show that there are regular semi simple elements x, y ∈ G, x of prime order, and |y| is divisible by at most two primes, such that (formula presented). In fact in all but four cases, y can be chosen to be of square-free order. Using this result, we prove an effective version of a previous result of Larsen, Shalev, and Tiep by showing that, given any integer m ≥ 1, if the order of a finite simple group S is at least (formula presented), then every element in S is a product of two mth powers. Furthermore, the verbal width of x^{m} on any finite simple group S is at most (formula presented). We also show that, given any two non-trivial words w1, w2, if G is a finite quasi simple group of large enough order, then (formula presented).

Original language | English (US) |
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Pages (from-to) | 1401-1430 |

Number of pages | 30 |

Journal | American Journal of Mathematics |

Volume | 137 |

Issue number | 5 |

State | Published - Oct 1 2015 |

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

*American Journal of Mathematics*,

*137*(5), 1401-1430.