For a group whose order is the product of two distinct primes, What is the order of the center of the group?

Question

I was solving some questions on group theory and I came across a problem something like this:

Let $G$ be a group of order 77. Then the order of the centre of the group is:

My attempt:

$77= 11\times7$. Since $7\nmid(11-1)$, thus we have $G$ to be abelian. Since $G$ is abelian $G=Z(G)$. Therefore $O(G)=O(Z(G))=77$.

But the solution provided a hint saying that "the order of the the centre of the group is order of the largest subgroup of $G$", which is the only normal subgroup of $G$ and hence the order of the centre of the group is $11$.

Can anybody correct me where and what am I missing in this question?

Answer 1

Prove the following lemma: if $G/Z(G)$ is cyclic, then $G$ is abelian.

In particular this shows that $Z(G)$ never has prime index. So if $|G|= pq$ this rules out the cases $|Z(G)| \in \{p,q\}$. So either $G$ is abelian or its center is trivial.

Both cases are possible. Here seems to be a construction of a nonabelian group of order $pq$, where $p< q$ and $q = 1 \pmod{p}$.

Answer 2

Theorem. A group of order $pq$ is either abelian or has trivial centre.

Proof. Suppose the theorem does not hold. Then there exists a group $G$ of order $pq$ whose centre has order $p$, say. Therefore, $G/Z(G)$ is cyclic of order $q$. However, if $G/Z(G)$ is cyclic then $G$ is abelian (see this old question for a proof), a contradiction. QED

On the other hand, this is overkill for the question at hand. As the OP points out, because $7\nmid(11-1)$ the group is abelian (essentially by the classification of groups of order $pq$). Therefore, if $|G|=77$ then $|Z(G)|=77$.

Answer 3

For $g\in G\setminus Z(G)$ it is $Z(G)<C_G(g)<G$ (strict inequalities). Therefore, the cases $|Z(G)|=p,q$ are ruled out just by Lagrange's theorem. And $|Z(G)|=1$ is ruled out by the Class Equation, if $p\nmid q-1$.