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Some infinite Fibonacci groups

Published online by Cambridge University Press:  20 January 2009

D. L. Johnson
D. L. Johnson
Affiliation:
University of Nottingham
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The Fibonacci groups are a special case of the following class of groups first studied by G. A. Miller (7). Given a natural number n, let θ be the automorphism of the free group F = 〈x1, …, xn |〉 of rank n which permutes the subscripts of the generators in accordance with the cycle (1, 2, …, n). Given a word w in F, let R be the smallest normal subgroup of F which contains w and is closed under θ. Then define Gn(w) = F/R and write An(w) for the derived factor group of Gn(w). Putting, for r ≦ 2, k ≦ 1,

with subscripts reduced modulo n, we obtain the groups F(r, n, k) studied in (1) and (2), while the F(r, n, 1) are the ordinary Fibonacci groups F(r, n) of (3), (5) and (6). To conform with earlier notation, we write A(r, n, k) and A(r, n) for the derived factor groups of F(r, n, k), and F(r, n) respectively.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 19 , Issue 3 , March 1975 , pp. 311 - 314
Copyright
Copyright © Edinburgh Mathematical Society 1975

References

REFERENCES

(1) Campbell, C. M. and Robertson, E. F., A note on a class of generalized Fibonacci groups (to appear).Google Scholar
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(7) Miller, G. A., Groups generated by n operators each of which is the product of the n-1 remaining ones, Amer. J. Math. 30 (1908), 93–98.CrossRefGoogle Scholar