2019 ROMANIAN MATHEMATICAL OLYMPIAD

Question

Let n be a positive integer, and let G be a finite group of order n. A function f: G G is a *pseudoendomorphism* if f(xyz) = f(x)f(y)f(z), for all x, y, z in G. a) If n is odd, show that every pseudoendomorphism of G is an endomorphism. b) If n is even, is every pseudoendomorphism of G an endomorphism?

Accepted Answer

a) Let e denote the unit of G. Let x = y = z = e to write f(e)^3 = f(e), so f(e)^2 = e. Since n is odd, it follows that f(e) = e. If x and y are members of G, write f(xy) = f(xye) = f(x)f(y)f(e) = f(x)f(y), to conclude that f is indeed an endomorphism of G. b) The answer is negative. Let a be an order 2 element of G, and let f: G G, f(x) = a. If x, y, z are elements of G, then f(xyz) = a = a^3 = f(x)f(y)f(z), so f is a pseudoendomorphism. However, f is not an endomorphism, since f(e) = a e.