5 Stating and proving important properties of homomorphisms

5) Stating and proving important properties of homomorphisms. (Give me some examples and solve them)

Solution

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.

note here ** is any random operation like +,-,*,/ etc.

examples

Let (G, ) be an arbitrary group and H = {e}, f(x y) = e = e e = f(x) f(y).
hence it is an homo morphism.