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A NOTE ON NIELSEN EQUIVALENCE IN FINITELY GENERATED ABELIAN GROUPS

Part of: Special aspects of infinite or finite groups

Published online by Cambridge University Press:  16 June 2011

DANIEL OANCEA
DANIEL OANCEA*
Affiliation:
1549 Victoria St. E, Whitby, Ontario, Canada L1N 9E3 (email: daniel.oancea@opg.com)
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Abstract

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Nielsen transformations determine the automorphisms of a free group of rank n, and also of a free abelian group of rank n, and furthermore the generating n-tuples of such groups form a single Nielsen equivalence class. For an arbitrary rank n group, the generating n-tuples may fall into several Nielsen classes. Diaconis and Graham [‘The graph of generating sets of an abelian group’, Colloq. Math.80 (1999), 31–38] determined the Nielsen classes for finite abelian groups. We extend their result to the case of infinite abelian groups.

Keywords

Nielsen transformationsabelian groups

MSC classification

Secondary: 20F99: None of the above, but in this section
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 84 , Issue 1 , August 2011 , pp. 127 - 136
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

References

[1]Coxeter, H. S. M. and Moser, W. O. J., Generators and Relations for Discrete Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, 14 (Springer, Berlin, 1972), pp. 85–92.CrossRefGoogle Scholar
[2]Diaconis, P. and Graham, R., ‘The graph of generating sets of an abelian group’, Colloq. Math. 80 (1999), 31–38.CrossRefGoogle Scholar
[3]Nielsen, J., ‘Die Isomorphismengruppe der Freien Gruppen’, Math. Ann. 91 (1924), 169–209.CrossRefGoogle Scholar