Hungerford's notation of $(|G|,|H|)=1$.

Question

I'm working on exercise p63,#5 in Hungerford's Algebra under Graduate Texts in Mathematics. It states that

"Let $G$, $H$ be finite cyclic groups. Then $G \times H$ is cyclic if and only if $(|G|,|H|) = 1$."

Can someone tell me what the $(|G|,|H|) = 1$ is? I can't get ahold of my professor during spring break. I also couldn't find the same notation in chapter 8. Thanks!!

Answer 1

It means that the greatest common divisor of the two quantities $|G|$ and $|H|$ is $1$.

In more detail: $(a,b)$ denotes the gcd of $a$ and $b$ and $|G|$ is the order of the group $G$.