## Abstract

In this paper we consider which families of finite simple groups G have the property that for each ∊ > 0 there exists N > 0 such that, if |G| ≥ N and S, T are normal subsets of G with at least ∊|G| elements each, then every non-trivial element of G is the product of an element of S and an element of T. We show that this holds in a strong and effective sense for finite simple groups of Lie type of bounded rank, while it does not hold for alternating groups or groups of the form PSLn(q) where q is fixed and n → ∞. However, in the case S = T and G alternating this holds with an explicit bound on N in terms of ∊. Related problems and applications are also discussed. In particular we show that, if w1, w2 are non-trivial words, G is a finite simple group of Lie type of bounded rank, and for g ∈ G, P_{w}_{1(}G_{),w2(}G_{)}(g) denotes the probability that g1g2 = g where gi ∈ wi(G) are chosen uniformly and independently, then, as |G| → ∞, the distribution P_{w}_{1(}G_{),w2(}G_{)} tends to the uniform distribution on G with respect to the L^{∞} norm.

Original language | American English |
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Pages (from-to) | 863-885 |

Number of pages | 23 |

Journal | Transactions of the American Mathematical Society |

Volume | 377 |

Issue number | 2 |

DOIs | |

State | Published - Feb 2024 |

### Bibliographical note

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