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One-relator groups that are residually of prime power order

Published online by Cambridge University Press:  09 April 2009

D. Gildenhuys
D. Gildenhuys
Affiliation:
Department of Mathematics, McGill University, Montreal 110, Canada
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If C is a class of groups, we denote by RC the class of groups which are residually in C i.e. G ∈ RC if and only if 1 ≠ g ∈ G implies that there exists a normal subgroup N of G such that g ∈ N and G/N ∈ C. A group G is residually a finite p-group if it belongs RFp, where Fp denotes the class of finite p-groups. One also says that the groups in RFp are residually of order equal to a power of the prime p. Given a group G with one defining relator r, one might ask for conditions on the “form” of the relator that would guarantee that G have certain residual properties. In this context, Baumslag (1971) has proved that if all the exponents of the generators appearing in r are positive, then G is residually solvable. In the same paper he also concerned himself with the residual nilpotence of one-relator groups, and found that the situation there was much more complicated. If one goes one step further and asks for conditions that will ensure that for a given prime p the one-relator group be residually a finite p-group, then very little seems to be known. Of course, if one takes r to be one of the generators: then G is freely generated by the remaining generators, and hence is in RFp for all primes p (Mahec (1949), Lazard (1965), 3.1.4). Our main purpose in this paper is to develop methods of generating examples of one-relator groups that are residually of order equal to a given prime p.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 19 , Issue 4 , June 1975 , pp. 385 - 409
Copyright
Copyright © Australian Mathematical Society 1975

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