FTFGAG: Fundamental Theorem of Finitely Generated Abelian Groups

Fundamental Theorem of Finitely Generated Abelian Groups

Primary decomposition

Every finitely generated abelian group G is isomorphic to a group of the form \displaystyle \mathbb{Z}^n\oplus\mathbb{Z}_{q_1}\oplus\dots\oplus\mathbb{Z}_{q_t} where n\geq 0 and q_1,\dots,q_t are powers of (not necessarily distinct) prime numbers. The values of n, q_1, \dots, q_t are (up to rearrangement) uniquely determined by G.

Invariant factor decomposition

We can also write G as a direct sum of the form \displaystyle \mathbb{Z}^n\oplus\mathbb{Z}_{k_1}\oplus\dots\oplus\mathbb{Z}_{k_u}, where k_1\mid k_2\mid k_3\mid\dots\mid k_u. Again the rank n and the invariant factors k_1,\dots,k_u are uniquely determined by G.

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