Group theory and generalizations Concept

Invariant factors and elementary divisors

Summary

Let $G$ be a finitely generated abelian group, and write $G \cong \mathbb{Z}^n \oplus \mathbb{Z}/k_1\mathbb{Z} \oplus \cdots \oplus \mathbb{Z}/k_u\mathbb{Z}$ with $1 < k_1 \mid k_2 \mid \cdots \mid k_u$ . The integers $k_1, \ldots, k_u$ are the invariant factors of $G$ . In the other form $G \cong \mathbb{Z}^n \oplus \mathbb{Z}/q_1\mathbb{Z} \oplus \cdots \oplus \mathbb{Z}/q_t\mathbb{Z}$ , the prime powers $q_1, \ldots, q_t$ are the elementary divisors of $G$ .

Remarks

The structure theorem says that the rank $n$ , the invariant factors and the elementary divisors are unique, and that they determine $G$ up to isomorphism. The elementary divisors are the prime power factors of the invariant factors. For example, the invariant factors $2, 12$ give the elementary divisors $2, 4, 3$ . For a finite group $G$ , the product of the invariant factors is $|G|$ , and the largest invariant factor $k_u$ is the exponent of $G$ . The same terms are used for finitely generated modules over a principal ideal domain.

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Tag	`0WHCA4SVP2`
URI	`m2://0WHCA4SVP2`
Kind	concept
Writing	W0 stub
MSC	20K01, 20K25

Invariant factors and elementary divisors

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