Hey,

I have this question which asks to prove that if a group has an element, a such that |a|=2, (exactly one order 2 element a) that this element is in the centre of the group.

so ag=ga for all g in G

aga^{-1}=g, aa^{-1}g=g so aa^{-1}=e which is true since a has order 2.

But I feel like this isn't enough at all and I cant really see where the "exactly 1 element with |a|=2" comes in,

Does anyone have any ideas?