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Coessential Abelianization Morphisms in the Category of Groups

Published online by Cambridge University Press:  20 November 2018

D. Oancea
D. Oancea*
Affiliation:
1549 Victoria St. E., Whitby, ON, L1N 9E3 e-mail: daniel.oancea@opg.com
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Abstract.

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An epimorphism $\phi :\,G\,\to \,H$ of groups, where $G$ has rank $n$, is called coessential if every (ordered) generating $n$-tuple of $H$ can be lifted along $\phi $ to a generating $n$-tuple for $G$. We discuss this property in the context of the category of groups, and establish a criterion for such a group $G$ to have the property that its abelianization epimorphism $G\,\to \,{G}/{[G,G]}\;$, where $[G,\,G]$ is the commutator subgroup, is coessential. We give an example of a family of 2-generator groups whose abelianization epimorphism is not coessential. This family also provides counterexamples to the generalized Andrews–Curtis conjecture.

Keywords

20F0520F9920J15coessential epimorphismNielsen transformationsAndrew-Curtis transformations
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 56 , Issue 2 , 01 June 2013 , pp. 395 - 399
Copyright
Copyright © Canadian Mathematical Society 2013

References

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