Groups

[kamil9876]

Let be a group with only one element of order 2, say . Prove that for all

[kamil9876]

finally solved it Want to see if there are more general solutions/better than mine/ remarks on general theory.

After some playing around:

I noticed that:

let x be any element in G:
by closure.


Because g is the only element of order 2. has only two solutions: 1 and g.

But because, for all then , namely

First case:


(1)

While the second case is impossible since it implies:


but g has order 2 so this cannot be the case. Hence (1) is the only case.