Random and Deterministic Triangle Generation of Three-Dimensional Classical Groups III

Abstract

Let p₁, p₂, p₃ be primes. This is the final paper in a series of three on the (p₁, p₂, p₃)-generation of the finite projective special unitary and linear groups PSU ₃(pⁿ), PSL ₃(pⁿ), where we say a noncyclic group is (p₁, p₂, p₃)-generated if it is a homomorphic image of the triangle group T_{p1, p2, p3} . This article is concerned with the case where p₁ = 2 and p₂ ≠ p₃. We determine for any primes p₂ ≠ p₃ the prime powers pⁿ such that PSU ₃(pⁿ) (respectively, PSL ₃(pⁿ)) is a quotient of T = T_{2, p2, p3} . We also derive the limit of the probability that a randomly chosen homomorphism in Hom(T, PSU ₃(pⁿ)) (respectively, Hom(T, PSL ₃(pⁿ))) is surjective as pⁿ tends to infinity.

Key Words:

• Classical groups
• Finite simple groups
• Triangle groups

2000 Mathematics Subject Classification:

• 20P05

ACKNOWLEDGMENTS

This article comprises part of my Ph.D. work which was supervised by Professor Martin Liebeck and supported financially by the Engineering and Physical Sciences Research Council (U.K.). I would like to especially thank Professor Liebeck for his help and encouragement.

Notes

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The integer n ₀ is defined so that U ₃(p ^{n ₀} ) is the smallest subfield subgroup of U ₃(p ⁿ ) containing elements of orders p ₂ and p ₃. q = p ⁿ and q ₀ = p ^{n ₀} . α, β, γ ∈ ℤ, α ≥0, β > 0 odd, γ > 0 odd such that (γ, 3) = 3 if α > 0. For i ∈ {2, 3}, if p ≠ p _i , then ℓ _i (q) is the order of q modulo p _i ; otherwise ℓ _i (q) = 1. For i ∈ {2, 3}, if p ≠ p _i , then ℓ _i (q ₀) is the order of q ₀ modulo p _i ; otherwise ℓ _i (q ₀) = 1. If p ₂ = 3, then ℓ = ℓ₃, ℓ(q) = ℓ₃(q), and ℓ(q ₀) = ℓ₃(q ₀).

The integer n ₀ is defined so that L ₃(p ^{n ₀} ) is the smallest subfield subgroup of L ₃(p ⁿ ) containing elements of orders p ₂ and p ₃. q = p ⁿ and q ₀ = p ^{n ₀} . α, β ∈ ℤ, α > 0, β > 0, (β, 6) ≠ 1. For i ∈ {2, 3}, if p ≠ p _i , then ℓ _i (q) is the order of q modulo p _i ; otherwise ℓ _i (q) = 1. For i ∈ {2, 3}, if p ≠ p _i , then ℓ _i (q ₀) is the order of q ₀ modulo p _i ; otherwise ℓ _i (q ₀) = 1. If p ₂ = 3, then ℓ = ℓ₃, ℓ(q) = ℓ₃(q), and ℓ(q ₀) = ℓ₃(q ₀).

Given primes p, p ₂, p ₃, we let e denote the number of positive integers n such that is a (p ₁, p ₂, p ₃)-group.

Communicated by P. Tiep.