This is a simple theorem I have read online:

With the following proof:Let G be a group and let $\displaystyle g, h \in G$ be commuting elements of finite orders m, n, respectively,

where $\displaystyle gcd(m,n)=1$. Then $\displaystyle (g) \cap (h)$ is the trivial subgroup.

The question is: What does "commuting element" mean? If it's the ordinary meaning (Like in ab=ba).Let $\displaystyle x \in (g) \cap (h)$. Then |x| is a divisor of |g| and of |h|, and so |x| is a divisor of

gcd(|g|,|h|)=1. Thus x is the identity of G.

What is the role in the proof and why cannot the above theorem be proven for non-commuting elements?