On triangle generation of finite groups of Lie type
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Claude Marion
Abstract
This paper is concerned with the (p1, p2, p3)-generation of finite groups of Lie type, where we say that a group is (p1, p2, p3)-generated if it is generated by two elements of orders p1, p2 having product of order p3. Given a triple (p1, p2, p3) of primes, we say that (p1, p2, p3) is rigid for a simple algebraic group G if the sum of the dimensions of the subvarieties of elements of orders dividing p1, p2, p3 in G is equal to 2 dimG. We conjecture that if (p1, p2, p3) is a rigid triple for G then given a prime p, there are only finitely many positive integers r such that the finite group G(pr) is a (p1, p2, p3)-group. We prove that the conjecture holds in many cases. Finally, we classify the rigid triples for simple algebraic groups. The conjecture together with this classification puts into context many results on Hurwitz (2, 3, 7)-generation in the literature, and motivates a new study of the (p1, p2, p3)-generation problem for certain finite groups of Lie type of low rank.
© de Gruyter 2010
Articles in the same Issue
- On triangle generation of finite groups of Lie type
- Hurwitz generation of the universal covering of Alt(n)
- A classification of certain finite double coset collections in the exceptional groups
- Recognizing SL2(q) in fusion systems
- Complete LR-structures on solvable Lie algebras
- Involutions and free pairs of bicyclic units in integral group rings
- Hypercentral groups with all subgroups subnormal
- On pro-S groups
- On groups acting on contractible spaces with stabilizers of prime-power order
Articles in the same Issue
- On triangle generation of finite groups of Lie type
- Hurwitz generation of the universal covering of Alt(n)
- A classification of certain finite double coset collections in the exceptional groups
- Recognizing SL2(q) in fusion systems
- Complete LR-structures on solvable Lie algebras
- Involutions and free pairs of bicyclic units in integral group rings
- Hypercentral groups with all subgroups subnormal
- On pro-S groups
- On groups acting on contractible spaces with stabilizers of prime-power order
