Homomorphisms and Isomorphisms

If (G, *) and (H, •) are groups, then a function f : G −→ H is a homomorphism if f (x * y) = f (x) • f (y) for all x, y ∈ G. Example: Let (G, *) be an arbitrary group and H = {e}, then the function f : G −→ H such that f (x) = e for any x ∈ G is a homomorphism. In fact, f (x * y) = e = e • e = f (x) • f (y). f (x) = x for any x ∈ G is a homomorphism. In fact, f (x * y) = x * y = f (x) * f (y). + 2 be a function such that f (x) = [0] if x is even [1] if x is odd. Then f is a homomor-phism. In fact, if x + y is even, then f (x + y) = [0] = f (x) + f (y). Similarly, if x + y is odd, then f (x + y) = [1] = f (x) + f (y). × =0 be a function such that f (M) = det M for any M ∈ GL(2, R).