I need to prove that if a group contains exactly one element with order 2, then that element is the center of the group. Here is how I formulated the problem: Let A be a group with an element c such that c^2 = 1 (i.e. c = c^-1), and b^2 = 1 implies b=c. Want either: ac = ca, for all a in A ca(c^-1) = a, for all a in A. After doing this, I tried manipulating the symbols for a while using different identities and properties that I know, but I can't seem to come up with any equations where one side doesn't reduce to the other trivially. I have a feeling that there is one small but crucial thing I am missing. Abelian groups have crossed my mind a few times, but I don't know how to use that. Does anyone have any suggestions for where to go from here?