## Abstract

Let G be a finite group, and let k be a nonnegative integer. We say that G has uniform spread k if there exists a fixed conjugacy class C in G with the property that for any k nontrivial elements x(1),...,x(k) in G there exists y is an element of C such that G = <x(i), y > for all i. Further, the exact uniform spread of G, denoted by u(G), is the largest k such that G has the uniform spread k property. By a theorem of Breuer, Guralnick, and Kantor, u(G) > 1 for every finite simple group G. Here we consider the uniform spread of almost simple linear groups. Our main theorem states that if G = <PSLn(q),g > is almost simple, then u(G) > 1 (unless G is isomorphic to S_6), and we determine precisely when u(G) tends to infinity as |G| tends to infinity.

Original language | English |
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Pages (from-to) | 35-109 |

Number of pages | 75 |

Journal | Nagoya Mathematical Journal |

Volume | 209 |

DOIs | |

Publication status | Published - 2013 |

## Keywords

- CHARACTERS
- FINITE CLASSICAL-GROUPS
- MAXIMAL-SUBGROUPS
- ORDER
- ELEMENTS
- FIXED-POINT RATIOS
- PROBABILISTIC GENERATION
- (2,3)-GENERATION