Solvable matrix groups and the Burnside problem
Abstract
Description
All groups are 2-generator. For any prime-power q, Theorem 1 constructs a solvable matrix group over a quotient of a Laurent polynomial ring. This group is closely related to a group of exponent q as shown in Theorems 2 & 3 . Theorem 4 in section 5 shows that a group of prime-power exponent contains the relations of a solvable group. It follows that the Burnside groups of exponent q are solvable, and it is easy to deduce that the solvability class of these groups tends to infinity with q. Crucial to this work, especially for precise bounds on the solvability class, is an earlier paper with Heilbronn and Mochizuki.
14 pages. Section 5, which is 3 pages, has been completely rewritten in an effort to make it simple and clear; no background, only an understanding of the commutative square on page 1 is needed. Sections 2 and 3 are the same as in earlier versions of this article which have been circulating. As these sections are considered correct, the proofs in these sections could be skipped in a first reading if time is a problem. Section 4 is new, but is not needed for section 5
14 pages. Section 5, which is 3 pages, has been completely rewritten in an effort to make it simple and clear; no background, only an understanding of the commutative square on page 1 is needed. Sections 2 and 3 are the same as in earlier versions of this article which have been circulating. As these sections are considered correct, the proofs in these sections could be skipped in a first reading if time is a problem. Section 4 is new, but is not needed for section 5