If $|G|=7 \times 11 \times 19$, then $G$ is abelian.

Question

If $|G|=7 \times 11 \times 19$,

then $G$ is abelian.

I have tried it by Sylow theorem but I failed.

Could someone give me the details?

Answer 1

Hint

Show that $$G\cong \mathbb Z/7\mathbb Z\times \mathbb Z/11\mathbb Z\times \mathbb Z/19\mathbb Z.$$

Notice that Sylow (or in fact Cauchy) tells you that there is an element of order $7$, an element of order $11$ and an other of order $19$.

It's almost finish.