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however, that if one defines

:

and

:

then one has in particular , , and

:

One can check that this yields the orders in the previous examples as special cases (see Hillar & Rhea).

Finitely generated abelian groups

An abelian group is finitely generated if it contains a finite set of elements (called generators) such that every element of the group is a linear combination with integer coefficients of elements of .

Let be a

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