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Coessential Abelianization Morphisms in the Category of Groups

Published:2011-09-15
Printed: Jun 2013
• D. Oancea,
1549 Victoria St. E. , Whitby, ON, L1N 9E3
 Format: LaTeX MathJax PDF

Abstract

An epimorphism $\phi\colon G\to H$ 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: coessential epimorphism, Nielsen transformations, Andrew-Curtis transformations
 MSC Classifications: 20F05 - Generators, relations, and presentations 20F99 - None of the above, but in this section 20J15 - Category of groups

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