Order (group Theory) - in Relation To Homomorphisms

In Relation To Homomorphisms

Group homomorphisms tend to reduce the orders of elements: if f: G → H is a homomorphism, and a is an element of G of finite order, then ord(f(a)) divides ord(a). If f is injective, then ord(f(a)) = ord(a). This can often be used to prove that there are no (injective) homomorphisms between two concretely given groups. (For example, there can be no nontrivial homomorphism h: S₃ → Z₅, because every number except zero in Z₅ has order 5, which does not divide the orders 1, 2, and 3 of elements in S₃.) A further consequence is that conjugate elements have the same order.