On the (2, 3, 7)-Generation of Some Special Linear Groups

ABSTRACT

In Lucchini et al. (2000), proved, in particular, that the groups SL _n (q) and SL _n (Z) are (2, 3, 7)-generated for each prime power q and each integer n ≥ 287, where Z is the ring of integers. Moreover, the method used in their article also applies for 93 smaller ranks less than 287, and our interest here is in finding other small ranks for which (2, 3, 7)-generation can be established. In this article, we find altogether 50 new ranks n for which the groups SL _n (q) and SL _n (Z) are (2, 3, 7)-generated.

Communicated by P. Higgins.

Keywords:

• (2, 3, 7)-generation
• Hurwitz group
• Special linear group

Mathematics Subject Classification:

• Primary 20F05
• Secondary 20G40

ACKNOWLEDGMENT

The author would like to express his deep gratitude to Professor John S. Wilson for his generous help and encouragement.