You're supposed to prove that for all not assume and play around with it (a common mistake with many students).
This is what you do. Let Show that order But is the unique element of order in What does this tell you?
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-1g=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?
So you can say that since a is in the group then its conjugate must also be in the group, let b be its conjugate then
bb=g-1agg-1ag= g-1aag=g-1g= e so b also had order 2
Since a is the only element of order to b must be a
Does that look alright?