Proving another group infinite.

Geoff C. Smith

School of Mathematical Sciences, University of Bath, Bath BA2 7AY, England

Let G = F (4, 7) denote the 7 generator group with 4-step Fibonacci defining relations. To be explicit, G has generators {g_{i} | i ∈ Z/7Z} and relators g_{i}g_{i+1}g_{i+2}g_{i+3}g_{i+4}^{−1} for each i ∈ Z/7Z. THEOREM The group G is infinite. The cardinality of this group was not hitherto known. The groups F (r, n) have been intensively studied, and many families are reasonably well understood. For example, F (2, n) is finite only when n is 1, 2, 3, 4, 5 or 7. For large n this was proved by Lyndon (unpublished) by using small cancellation theory. For very small n the results can be obtained by hand using ingenuity or by the Todd-Coxeter coset enumeration process. The latter technique is best left to a machine. For moderate values of n the issue was resolved by Brunner for 8 and 10, and for n = 9 by M.F. Newman [7] using an ingenious embellishment of the Golod-Šaferevič theorem [4]. Various families F (r, n) with r > 2 are of known cardinality, and something of the structure of some of the finite ones is understood [1]. The literature is extensive, but we draw the reader’s attention the survey article of Thomas [9] which gives an excellent description of the field. The group studied in this paper does not fall into these understood families. We will show that G is infinite by appeal to a slightly strengthened form of the Golod-Šaferevič theorem. We state it now, in the version which appears in [6] (given the remarks in section 15.3 of that work). THEOREM (Golod-Šaferevič) Let G =< X | R > be a finite group with |X| = d and |R| = r. Suppose that p is a rational prime with the property that G/G′G^{p} has order p^{d}, then r > d^{2}/4. The proof of our theorem depends on finding a subgroup of G to which we may apply the Golod-Šaferevič theorem. Such a subgroup was found with the help of both CAYLEY [2] and QUOTPIC [5]. The former is a well-known Computational Group Theory system. The latter is a new group theory graphics system created by Holt and Rees. Let M =< a, b | a^{3}, b^{7}, b^{a} = b^{2} > be a non-abelian group of order 21. Notice that M has an automorphism ϕ of order 7 which fixes b and sends a to ab^{3}. A routine calculation yields that a(a)ϕ(a)ϕ^{2}(a)ϕ^{3} = (a)ϕ^{4}. We may apply powers of ϕ to this equation to deduce that there is an epimorphism θ : G ⟶ M defined by (g_{i})θ = (a)ϕ^{i−1}. The kernel H of θ can be shown to have a presentation on 12 generators and 32 relators. This can be accomplished via a computer program which uses the Reidemeister-Schreier method and Tietze transformations. One can read off from this presentation that H/H^{2} has order 2^{12}. Hence, by the Theorem of Golod and Šaferevič [6], the group H is infinite. These calculations were independently checked by S. Rees using code of D. F. Holt, and by M. F. Newman using SPAS [3], a product of Neubüser’s school at Aachen. The SPAS system includes many stand-alone programs written by G. Havas and others. Added in revision D .B. Nikolova and E. F. Robertson [8] have recently used related techniques to prove that F (8, 10) is infinite.

References

[1] C. M. CAMPBELL and E. F. ROBERTSON, ‘The orders of certain metacyclic groups’, Bull. London Math. Soc. 6 (1974), 312-314.

[2] J. J. CANNON, ‘An introduction to the group theory language Cayley’ in Computational Group Theory, M.D. Atkinson (ed.), Academic Press, London, 1984, 145–183.

[3] V. FELSCH et al. , ‘User’s Reference Manual, SPAS 2.5’, Lehrstuhl D für Mathematik, RWTH Aachen, (1989).

[4] W. GASCHÜTZ and M. F. NEWMAN, ‘On presentations of finite p−groups’, J. Reine Angew. Math. 245 (1970), 172-176.

[5] D. F. HOLT and S. REES, ‘A graphics system for displaying finite quotients of finitely presented groups’ in Groups and Computation, Larry Finkelstein and William M. Kantor (eds.), DIMACS Series in Discrete Mathematics and Theoretical Computer Science 11, AMS 1993, 113-126.

[6] D. L. JOHNSON, Presentations of Groups, 1990, CUP.

[7] M. F. NEWMAN, ‘Proving a Group Infinite’, Arch. Math.(Basel) 54 (1990) No. 3, 209-211.

[8] D. B. NIKOLOVA and E. F. ROBERTSON, ‘One more infinite Fibonacci group’, C. R. Acad. Bulgare Sci 46 (1993) No. 3 13-15.

[9] R. M. THOMAS, ‘The Fibonacci Groups Revisited’, in C. M. Campbell and E. F. Robertson (eds.), Proceedings of Groups St. Andrews 1989 vol. 2 LMS Lecture Note Series 160, CUP, (1991) pp 445-454.