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Finitely generated abelian group

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= Commutative group
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= Structure theorem for finitely generated abelian groups
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Every finitely generated <abelian group> is isomorphic to
$$
\mathbb Z^r\oplus\mathbb Z/d_1\mathbb Z\oplus\cdots\oplus\mathbb Z/d_s\mathbb Z,
\qquad d_1\mid d_2\mid\cdots\mid d_s.
$$
The rank $r$ and invariant factors $d_i>1$ are unique.

Fundamental theorem of finitely generated abelian groups

Finite abelian group

Finite generation detected by a prime quotient

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Fundamental theorem of finitely generated abelian groups

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Fundamental theorem of finitely generated abelian groups

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