Centre of a group

Unique group involutions are central

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

Proof

Let be the unique involution in group . Let some and . Then , thus . The first case immediately implies , a contradiction, therefore . Hence and moreover for arbitrary , wherefore .

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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