Group theory and generalizations Abelian groups Theorem

Classification of finite abelian groups

Builds on Abelian group Cyclic group Group isomorphism Direct product of groups Product of cyclic groups is cyclic iff the orders are coprime

Statement

Let $G$ be a finite abelian group. Then $G$ is isomorphic to a direct sum $\mathbb{Z}/q_1\mathbb{Z} \oplus \cdots \oplus \mathbb{Z}/q_t\mathbb{Z}$ , where each $q_i$ is a power of a prime number. The numbers $q_1, \ldots, q_t$ are unique up to their order. Equivalently, $G \cong \mathbb{Z}/k_1\mathbb{Z} \oplus \cdots \oplus \mathbb{Z}/k_u\mathbb{Z}$ with $1 < k_1 \mid k_2 \mid \cdots \mid k_u$ , and these $k_i$ are unique.

Remarks

The two forms are equivalent because $\mathbb{Z}/m n\mathbb{Z} \cong \mathbb{Z}/m\mathbb{Z} \oplus \mathbb{Z}/n\mathbb{Z}$ if and only if $\gcd(m, n) = 1$ , as in this statement. Each abelian group of order $8$ is isomorphic to exactly one of $\mathbb{Z}/8\mathbb{Z}$ , $\mathbb{Z}/4\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$ and $\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$ . Each abelian group of order $15$ is cyclic, because $15 = 3 \cdot 5$ with $\gcd(3, 5) = 1$ . The theorem is the case of rank $0$ of the structure theorem for finitely generated abelian groups.

History

Leopold Kronecker proved the theorem in 1870 with a group-theoretic proof, but he did not state it in the language of groups. His work extended an earlier result of Carl Friedrich Gauss from 1801 on quadratic forms. Ferdinand Georg Frobenius and Ludwig Stickelberger stated and proved the theorem in the language of groups in 1878.

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To understand this page from the ground up, read these 18 pages in order. The foundations come first.

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Binary operation definition
Group concept
Commutativity definition
Direct product of groups definition
Group homomorphism definition
Laws of exponents in a group statement
Order of a group definition
Subgroup definition
Abelian group definition
Group isomorphism definition
Order of an element definition
Cyclic subgroup generated by an element definition
g to the n equals 1 iff the order divides n statement
Cyclic group definition
Order of a cyclic subgroup equals the order of its generator statement
Cyclic groups of the same order are isomorphic statement
Product of cyclic groups is cyclic iff the orders are coprime statement
Classification of finite abelian groups statement

Where this leads

7 pages build on this one.

Web https://en.wikipedia.org/wiki/Abelian_group, Classification
Web https://en.wikipedia.org/wiki/Finitely_generated_abelian_group, Primary decomposition
Web https://en.wikipedia.org/wiki/Finitely_generated_abelian_group, Invariant factor decomposition
Web https://en.wikipedia.org/wiki/Finitely_generated_abelian_group, History

About this page

Tag	`07CQR9BW3R`
URI	`m2://07CQR9BW3R`
Aliases	`fundamental-theorem-of-finite-abelian-groups`, `basis-theorem-for-finite-abelian-groups`
Kind	statement, theorem
Writing	W0 stub
MSC	20K01
Status	This copy of the site has no build data. Run `m2 status 07CQR9BW3R`.
Uses	Abelian group, Cyclic group, Group isomorphism, Direct product of groups, Product of cyclic groups is cyclic iff the orders are coprime

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