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The soluble subgroups of a one-relator group with torsion

Published online by Cambridge University Press:  09 April 2009

B. B. Newman
B. B. Newman
Affiliation:
James Cook UniversityTownsville, Queensland, 4810, Australia
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In a survey article [1] Baumslag posed the problem of determining the abelian subgroups of a one-relator group. The solution of this problem was stated but not proved in [5], and partly solved by Moldavanskii [4]. In this paper it will be proved that the centralizer of every non-trivial element in a one-relator group with torsion is cyclic, and that the soluble subgroups of a one-relator groups with torsion are cyclic groups or the infinite dihedral group. That both types of groups may occur as subgroups is easily seen by considering

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 16 , Issue 3 , November 1973 , pp. 278 - 285
Copyright
Copyright © Australian Mathematical Society 1973

References

[1]Baumslag, G., ‘Groups with one defining relator’, J. Austral. Math. Soc., 4 (1964), 385–392.CrossRefGoogle Scholar
[2]Lewin, T., ‘Finitely generated D-groups’, J. Austral. Math. Soc., 7 (1967), 375–409.CrossRefGoogle Scholar
[3]Magnus, W., Karrass, A., and Solitar, D., Combinatorial Group Theory (Interscience, 1966).Google Scholar
[4]Moldavanskii, D. I., ‘Certain subgroup of groups with one defining relation’, (Russian), Sibirsk. Mat. Z. 8 (1967), 1370–1384.Google Scholar
[5]Newman, B. B., ‘Some results on one-relator groups’,Bull. Amer. Math. Soc., 74 (1968), 568–571.CrossRefGoogle Scholar