Question

Prove that Z_m x Z_nis cyclic if and only if (m,n) = 1.

Short answer

It has been proved that Z_m x Z_nis cyclic if and only if (m,n)= 1

Step-by-step solution

Proof of the ‘only if’ part

Let us suppose that $Z_m\times Z_n$is cyclic.

To Prove: (m,n) = 1

Proof: If $(m,n)\ne 1,then(m,n)=d>1$for some integer d.

This implies that ddivides mand ddivides n.

Thus, ddivides mn.

This implies $p=\frac{mn}{d}$is an integer.

Thus, we can see that every element of $Z_m\times Z_n$has an order dividing p.

This implies $Z_m\times Z_n$is not cyclic.

This contradicts our supposition.

Hence, (m,n) = 1.

Proof of the ‘if’ part

Let (m,n)=1

To prove: $Z_m\times Z_n$is cyclic.

Proof: Let a and b be generators of Z_mand Z_n , respectively.

This implies ‘order of a’ = m and ‘order of b’=n

This implies ‘order of ab’ = mn

This implies (a,b) is a generator of $Z_m\times Z_n$.

Hence, $Z_m\times Z_n$is cyclic.