Finite abelian group with specified invariant factors

Let $r\geq0$ , and for $r>0$ let $2\leq d_1\mid d_2\mid\cdots\mid d_r$ be integers. The finite abelian group with these invariant factors is the direct product

C_{d_1}\times\cdots\times C_{d_r},

where each factor is the additive group of residue classes modulo $d_i$ . The empty product for $r=0$ is the trivial group.

Classification and order

Its order is $\prod_{i=1}^r d_i$ , with empty product $1$ . The classification of finite abelian groups says that every finite abelian group occurs up to isomorphism, with a unique such list of invariant factors.

Simplicity

For $r=0$ the group is trivial. For $r\geq2$ , a coordinate factor is proper nontrivial normal. For $r=1$ , the cyclic group $C_{d_1}$ is simple exactly when $d_1$ is prime.

References

J. S. Milne, Group Theory, v4.01, Theorem 1.57, p. 26: invariant-factor classification of finite abelian groups.