J Austral Math Soc Ser A 53 pp352--368, 1992.

Finite One-Relator Products of Two Cyclic Groups with the Relator of Arbitrary Length

C. M. Campbell, P. M. Heggie, E. F. Robertson and R. M. Thomas

(Received 7 Spetember 1990; revised 1 February 1991)

Abstract

In this paper we consider the groups G = G(a, n) defined by the presentations

áa, b:a² = bⁿ = ab^-1ab(abab^-1)^a-1ab²ab^-2 = 1ñ.

We derive a formula for [G¢: G¢¢] and determine the order of G whenever n £ 7. We show that G is a finite soluble group if n is odd, but that G can be infinite when n is even, n ³ 8. We also show that G(6, 10) is a finite insoluble groups involving PSU(3, 4), and that the group H with presentation

áa, b:a² = b¹⁰, ab^-1ab(abab^-1)⁵ab²ab^-2 = 1ñ.

is a finite group of defiency zero of order at least 114,967,210,176,000.

1991 AMS Subject Classification: 20F05

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Authors

C. M. Campbell
P. M. Heggie
E. F. Robertson: Mathematical Institute, University of St Andrews, St Andrews KY16 9SS, Scotland.
R. M. Thomas: Department of Computing Studies, University of Leicester LE1 7RH, England.

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Last Modified: Mon Feb 3 9:46:11 2003