1. The problem statement, all variables and given/known data *Attached is the problem statement, along with a definition which is to be used (I feel like some of the definitions in this text are somewhat untraditional, so I am including this one for clarity). Edit* sorry, it's problem #31. 2. Relevant equations 3. The attempt at a solution Let <T, !> be any binary structure isomorphic with <S,*>. This means there is some bijection Ψ: S → T such that (∀s1,s2 ∈ S)(Ψ(s1*s2) = Ψ(s1)!Ψ(s2)). Now let d ∈ T. Since Ψ is surjective, there is some c ∈ S such that Ψ(c) = d. But then there is some x ∈ S such that x*x = c. As a result, Ψ(x*x) = Ψ(c) = d = Ψ(x)!Ψ(x). Q.E.D.