[SOLVED] Herstein algebra hint(s) desired 1. The problem statement, all variables and given/known data Let G be an abelian group of order n (even), [itex]a_1[/itex] ,..., [itex]a_n[/itex] its elements. Let x = [itex]a_1[/itex] [itex] \cdot[/itex] ... [itex]\cdot[/itex] [itex]a_n[/itex]. Show that if G has more than one element b which is not the identity e, such that [itex]b^2 = e[/itex], then x = e. (from Herstein, Abstract Algebra, section 2.4 problem 43b) 2. Relevant equations We know that [itex]x^2 = e[/itex]. 3. The attempt at a solution We can assume that b = [itex]a_1[/itex] , [itex]a_2[/itex] and note that we must also have a third solution b = [itex]a_1 \cdot \ a_2[/itex] as well which differs from the first two. It would seem that a reasonable strategy would be to attempt to find a solution to cx = c, but I'm afraid I don't see it (and not for lack of experimentation).