Therefore there are finitely generated groups that cannot be recursively presented.

Taking yields the fundamental theorem of finitely generated abelian groups.

Another important idea in geometric group theory is to consider finitely generated groups themselves as geometric objects.

Formally, this means classifying finitely generated groups with their word metric up to quasi-isometry.

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By contrast, classification of general infinitely generated abelian groups is far from complete.

Finitely generated abelian groups are completely classified and are particularly easy to work with.

This is the only way in which mobility on the labour market will not generate socially excluded groups, whose behaviour becomes deviant sooner or later.

It is much more difficult to construct finitely generated infinite simple groups.