If G1 and G2 are cyclic groups of orders m and n respectivel

If G_1 and G_2 are cyclic groups of orders m and n, respectively, prove that G_1 times G_2 is cyclic if and only if m and n are relatively prime.

Solution

Let |G₁| = m and |G₂| = n, so |G₁ G₂| = mn.

Assume G₁ G₂ is cyclic. Show the orders are relatively prime.

Let d = gcd (m, n) and let (g, h) be a generator for G₁ G₂. |(g, h)| = mn.

Consider (g, h)^mn/d = ((g^m) ^n/d , (h ⁿ)^m/d) = (e, e).

Then mn = |(g, h)| mn/d, so d = 1.

Conversely, suppose m and n are relatively prime.

We’ll show G₁ G₂ is cyclic. Choose generators g forG₁ and h for G₂.

That is, G₁ = < g > and G₂ = < h >.

Since gcd (m, n) = 1, lcm (m, n) = mn.

Then by well known Theorem, |(g, h)| = lcm(m, n) = mn = |G₁ G₂|,

so G₁ G₂ is cyclic.