Unique group involutions are central

If a group has a single unique involution (element $𝑎$ of order $| 𝑎 | = 2$ ) then that involution is in the group’s centre, i.e. it commutes with every element $𝑥$ so that $𝑎 𝑥 = 𝑥 𝑎$ . group

Proof

Let $𝑎 = 𝑎 - 1$ be the unique involution in group $𝐺$ . Let some $𝑥 \in 𝐺$ and $𝑦 = 𝑥 𝑎 𝑥 - 1$ . Then $𝑦 2 = 𝑥 𝑎 𝑥 - 1 𝑥 𝑎 𝑥 - 1 = 𝑒$ , thus $𝑦 \in {𝑒, 𝑎}$ . The first case immediately implies $𝑎 = 𝑒$ , a contradiction, therefore $𝑦 = 𝑎$ . Hence $𝑎 = 𝑥 𝑎 𝑥 - 1$ and moreover $𝑎 𝑥 = 𝑥 𝑎$ for arbitrary $𝑥 \in 𝐺$ , wherefore $𝑎 \in 𝑍 (𝐺)$ .

From the above proof it is clear this statement extends to any order.

Question

According to this StackExchange answer this is an example of a broader phenomenon where All group elements with a unique property are central. This is currently beyond my scope.

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Unique group involutions are central

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