This appears to be new to MSE.
I'm reading "Contemporary Abstract Algebra (Eighth Edition)," by Gallian.
This is Exercise 7.38 ibid. Answers that use only the tools available in the textbook so far will be preferred.
Prove that if $G$ is a finite group, then the index of $Z(G)$ cannot be prime.
Thoughts:
What I had in mind first of all was to try the contrapositive:
Let $G$ be a group. Then if $[G:Z(G)]=p$ for some prime $p$, then $G$ is infinite.
No progress worth sharing was made after a short while here.
Next, I considered when $G=Z(G)$; that is, when $G$ is abelian; with $G$ finite. Then $[G:Z(G)]=1$ is not prime. This does not seem helpful.
The section of the textbook ibid. this exercise is from is on "Cosets and Lagrange's Theorem". This gives me some indication that an argument at the "coset level/definition" of the index of a group would be most appropriate.
None of the previous exercises leap out at me as helpful.
Please help :)