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

Abstract

Let p ₁, p ₂, p ₃ be primes. This is the second article in a series of three on the (p ₁, p ₂, p ₃)-generation of the finite projective special unitary and linear groups PSU₃(p ⁿ ), PSL₃(p ⁿ ), where we say a noncyclic group is (p ₁, p ₂, p ₃)-generated if it is a homomorphic image of the triangle group T _{p 1, p 2, p 3} . This paper is concerned with the case where p ₁ = 2 and p ₂ = p ₃. We determine for any prime p ₂ the prime powers p ⁿ such that PSU₃(p ⁿ ) (respectively, PSL₃(p ⁿ )) is a quotient of T = T _{2, p 2, p 2} . We also derive the limit of the probability that a randomly chosen homomorphism in Hom(T, PSU₃(p ⁿ )) (respectively, Hom(T, PSL₃(p ⁿ ))) is surjective as p ⁿ tends to infinity.

Key Words:

• Classical groups
• Finite simple groups
• Triangle groups

2000 Mathematics Subject Classification:

• 20P05

ACKNOWLEDGMENT

This paper 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

For i ∈ {1, 2, 3}, if C _i  = C _j for some j ∈ {4, 7, 8}, then .

For i ∈ {2, 3}, if C _i  = C ₆, then .

Conditions (*), (**), (+), (+ +), (−), (– –), (.), (:) are given in Subsection 3.1.

.

((V ₁, W ₁),…, (V _k , W _k )) is a k-tuple of distinct pairs of subspaces of 𝒱 where dimV _i  = 1 and dimW _i  = 2 for 1 ≤ i ≤ k.

m _V is the number of distinct V _i , m _W is the number of distinct W _i .

{V _{a 1} ,…, V _{a m V} } = {V _i : 1 ≤ i ≤ k}.

{W _{b 1} ,…, W _{b m W} } = {W _i : 1 ≤ i ≤ k}.

For two finite groups H, J, we write H ∼ J if |Hom*(T, H)| = |Hom*(T, J)|.

((V ₁, W ₁),…, (V _k , W _k )) is a k-tuple of distinct pairs of subspaces of 𝒱 with dimV _i  = 1 and dimW _i  = 2 for 1 ≤ i ≤ k.

m _V is the number of distinct V _i , m _W is the number of distinct W _i .

V ₁,…, V _{m V} are assumed to be distinct.

W ₁,…, W _{m W} are assumed to be distinct.

The groups G ₁, G ₂, J ₁, J ₂, H ₁ and H ₂ are defined in Subsection 6.1.

For two finite groups H, J, we write H ∼ J if |Hom*(T, H)| = |Hom*(T, J)|.

Here r = q − 1 and s = q + 1.

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 order p ₂.

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.

If p ≠ p ₂, then ℓ(q) is the order of q modulo p ₂; otherwise ℓ(q) = 1.

If p ≠ p ₂, then ℓ(q ₀) is the order of q ₀ modulo p ₂; otherwise ℓ(q ₀) = 1.

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 order p ₂.

q = p ⁿ and q ₀ = p ^{n ₀} .

α, β ∈ ℤ, α > 0, β > 0, (β, 6) ≠ 1.

If p ≠ p ₂, then ℓ(q) is the order of q modulo p ₂; otherwise ℓ(q) = 1.

If p ≠ p ₂, then ℓ(q ₀) is the order of q ₀ modulo p ₂; otherwise ℓ(q ₀) = 1.

Communicated by P. Tiep.