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The Abelian Case of Solitar's Conjecture on Infinite Nielsen Transformations

Published online by Cambridge University Press:  20 November 2018

Olga Macedonska-Nosalska*
Affiliation:
Instytut Matematyki, Politechnika Śalaska Zwyciȩstwa 42 44-100 gliwice, Poland
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Abstract

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The paper proves that the group of infinite bounded Nielsen transformations is generated by elementary simultaneous Nielsen transformations modulo the subgroup of those transformations which are equivalent to the identical transformation while acting in a free abelian group. This can be formulated somewhat differently: the group of bounded automorphisms of a free abelian group of countably infinite rank is generated by the elementary simultaneous automorphisms. This proves D. Solitar's conjecture for the abelian case.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1985

References

1. Cohen, R., Classes of automorphisms of free groups of infinite rank, Trans, of Amer. Math. Soc. 177, March 1973, pp. 99120.Google Scholar
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3. Macedonska-Nosalska, O., On infinite Nielsen transformations, Math. Scandinavica, 51 (1982), pp. 6386.Google Scholar
4. Magnus, W., Karrass, A. and Solitar, D., Combinatorial group theory, Interscience, New York, 1966.Google Scholar
5. Nielsen, J., Om Regnig med ikke-kommutative Faktorer og dens Anvendeise i Gruppeteorien, Mat. Tidsskr. B, 1921, pp. 7794.Google Scholar
6. Nielsen, J., Oie Isomorphismengruppe der frein Gruppen, Math. Ann. 91 (1924), pp. 169209.Google Scholar
7. Rado, R., A proof of the basis theorem for finitely generated abelian groups, J. London Math. Soc, 26 (1951), pp. 7475.Google Scholar
8. Whitehead, J. H. C., On certain sets of elements in a free group, Proc. London Math. Soc. 41 (1936), pp. 4856.Google Scholar
9. Whitehead, J. H. C. On equivalent sets of elements in a free group, Ann. of Math. 37 (1936), pp. 782—800.Google Scholar
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