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.

Explanations

L1 human

There are two standard ways to write a finitely generated abelian group as a sum of cyclic groups. The invariant factors are orders that divide each other. The elementary divisors are prime power orders.