Let φ be a homomorphism from a group G to a group and g be an element of G

Let φ be a homomorphism from a group G to a group and g be an element of G. Then

(1) φ carries the identity of G to the identity of .

(2) φ(gⁿ) = [φ(g)]ⁿ for all n in Z.

(3) If |g| is finite, then |φ(g)| divides |g|.

(4) Ker φ is a subgroup of G.

(5) If φ(g) = h, then .