Find the order of $Z(G)$.

Question

If $G$ be a group of order $pq$ where $p$ and $q$ are prime integers then find $|Z(G)|$. The options are i) 1 or $p$ ii) 1 or $q$ iii) 1 or $pq$ iv) None of these. I know groups of prime order are cyclic so $G$ will atleast have one subgroup of order $p$ or $q$. But how will I find out $|Z(G)|$ from this?

Answer 1

You will need the result that if $G/Z(G)$ is cyclic, then $G$ is abelian. This rules out $\lvert Z(G) \rvert=p$ or $q$. If $G$ is abelian, then $G=Z(G)$. Also, $\lvert Z(G) \rvert=1$ can be realized (an example is $G=S_3$, but can be generalized to $p$ or $q$ arbitrary, with $p \neq q$).