The orders of the Fibonacci groups

Abstract

SynopsisThe Fibonacci groups F(r, n) have been studied by various authors, chiefly in order to determine which ones are finite. This article contains a summary of the known results about this problem, followed by some further results obtained by the author. In particular, the orders of the groups F(r, 3) for r ≡ 2 (mod 3) and F(r, 4) for r ≡ 2 (mod 4) are determined, and various other Fibonacci groups are proved infinite by methods similar to those of Chalk and Johnson.