Old Q with less useful answers, new Q with useful answer. I thought of flagging the old Q as duplicate of the new one, but perhaps it's best to merge the new answer into the old thread.

@ccorn Are they really duplicates? There is a difference that one of them assumes $Z(G)\ne\{e\}$.

(Thinking again... You have a point there. Merging the new answer in the old thread would not fit perfectly.)

@ccorn But you are right that $Z(G)\ne\{e\}$ is mentioned in the question. Maybe it would be good to get moderator's input. (Since a mod will be needed for merging anyway.)

@DanielFischer Do you happen to be around. (Well, you are the only mod currently pingable in this room.)

@MartinSleziak Since the older question is a proof-verification question, merging wouldn't be appropriate. Closing either as a duplicate of the other would work. Not sure which direction would be better, the old question has a lot of views. Aside: Why don't more people give the argument in the direct order, since $G$ isn't abelian, $G/Z(G)$ is not cyclic, so the order of the centre can't be larger than $p$?

@DanielFischer Well, since it is not tagged proof-verification, perhaps it could be considered as a general question.

@ccorn See Daniel Fischer's response above. He suggest closing as a duplicate. Since you brought up the issue, perhaps you will have some suggestions about the direction of duplicate closure.

Thanks, Martin and Daniel. Dup-closing seems suboptimal as well: The older has the better question; the newer Q is clearly deficient but has the most upvoted (and IMHO useful) answer. I'll just add another link to the comment section of the old Q for now.

NB: Daniel's aside reminds me of times some years ago where I used to lurk in the main math chatroom, silently enjoying the occasions where Daniel would offer a proof, live, yet perfectly ordered and free from clutter. Nowadays I do not visit chatrooms often, but here I have done it again, and what happens? Yeah :-)