Abstract Algebra: Theory and Applications · Problem 9.4.49

Normal Subgroups and Factor Groups — Thomas Judson

Mathematics — Abstract Algebra · direct-products-cyclic

Problem

Let

n_1, \ldots, n_k

be positive integers. Show that

\prod_{i=1}^{k} \mathbb{Z}_{n_i} \cong \mathbb{Z}_{n_1 \cdots n_k}

if and only if

\gcd(n_i, n_j) = 1

for

i \neq j