Group Direct Product of Cyclic Groups/Corollary

Theorem

Let $n_1, n_2, \ldots, n_s$ be a finite sequence of integers, all greater than $1$, such that for any pair of them $n_i$ and $n_j$, $n_i \perp n_j$.

Let $G_i$ be a cyclic group of order $n_i$ for each $i: 1 \le i \le s$.

Then $G_1 \times G_2 \times \cdots \times G_s$ is cyclic of order $n_1 n_2 \ldots n_s$.

Proof

When $s = 1$ the result is trivial.

Assume the result holds for $s = k$.

Then $H = G_1 \times G_2 \times \ldots \times G_k$ is cyclic of order $n_1 n_2 \ldots n_k$.

Applying the main result to $H \times G_{k + 1}$ gives us the result for $s = k + 1$.

The result follows by induction.

$\blacksquare$

Sources

• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $13$: Direct products: Corollary $13.2$