The isomorphism problem for one-relator groups with non-trivial centre

1974 ◽  
Vol 136 (2) ◽  
pp. 95-106 ◽  
Author(s):  
Alfred Pietrowski
Keyword(s):  
Isomorphism Problem ◽  
One Relator Groups ◽  
Trivial Centre

On the isomorphism problem for one-relator groups

1976 ◽  
Vol 27 (1) ◽  
pp. 484-488 ◽  
Author(s):  
Gerhard Rosenberger ◽  
R. N. Kalia
Keyword(s):  
Isomorphism Problem ◽  
One Relator Groups

The isomorphism problem for a class of one-relator groups

1975 ◽  
Vol 217 (1) ◽  
pp. 53-57 ◽  
Author(s):  
Stephen Meskin
Keyword(s):  
Isomorphism Problem ◽  
One Relator Groups

scholarly journals A class of one-relator groups with centre

1991 ◽  
Vol 44 (2) ◽  
pp. 245-252 ◽  
Author(s):  
James McCool
Keyword(s):  
Two Stage ◽  
One Relator Groups ◽  
Trivial Centre ◽  
Repeated Applications ◽  
Suitable Group

Let the group H have presentation where m ≥ 3, pi ≥ 2 and (pi, pj) = 1 if i ≠ j. We show that H is a one-relator group precisely if H can be obtained from a suitable group 〈a, b; ap = bp〉 by repeated applications of a (two-stage) procedure consisting of applying central Nielsen transformations followed by adjoining a root of a generator. We conjecture that any one-relator group G with non-trivial centre and G/G′ not free abelian of rank two can be obtained in the same way from a suitable group 〈a, b; ap = bp〉.

scholarly journals The isomorphism problem for a class of para-free groups

1997 ◽  
Vol 40 (3) ◽  
pp. 541-549 ◽  
Author(s):  
Benjamin Fine ◽  
Gerhard Rosenberger ◽  
Michael Stille
Keyword(s):  
Group Theory ◽  
Free Group ◽  
Special Interest ◽  
Isomorphism Problem ◽  
Combinatorial Group Theory ◽  
History Of ◽  
One Relator Groups ◽  
Rank 2 ◽  
Combinatorial Group ◽  
Torsion Free

In 1962 Gilbert Baumslag introduced the class of groups Gi, j for natural numbers i, j, defined by the presentations Gi, j = < a, b, t; a−1 = [bi, a] [bj, t] >. This class is of special interest since the groups are para-free, that is they share many properties with the free group F of rank 2.Magnus and Chandler in their History of Combinatorial Group Theory mention the class Gi, j to demonstrate the difficulty of the isomorphism problem for torsion-free one-relator groups. They remark that as of 1980 there was no proof showing that any of the groups Gi, j are non-isomorphic. S. Liriano in 1993 using representations of Gi, j into PSL(2, pk), k ∈ ℕ, showed that G1,1 and G30,30 are non-isomorphic. In this paper we extend these results to prove that the isomorphism problem for Gi, 1, i ∈ ℕ is solvable, that is it can be decided algorithmically in finitely many steps whether or not an arbitrary one-relator group is isomorphic to Gi, 1. Further we show that Gi, 1 ≇ G1, 1 for all i > 1 and if i, k are primes then Gi, 1 ≅ Gk, 1 if and only if i = k.

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