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- Title
On triangle generation of finite groups of Lie type.
- Authors
Marion, Claude
- 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 dim G. 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.
- Subjects
FINITE groups; LIE groups; ALGEBRA; LOGICAL prediction; PRIME numbers
- Publication
Journal of Group Theory, 2010, Vol 13, Issue 5, p619
- ISSN
1433-5883
- Publication type
Article
- DOI
10.1515/JGT.2010.014