Let
φ be a homomorphism from a group G
to a group 
and H be a subgroup of G. Then
(1) If |H| = n, then |φ(H)| divides n.
(2) If 
, then 
.
(3) If 
, then 
.
(4) If φ is onto and Ker φ = {e}, then φ is an isomorphism from G
to .
Let
φ be a homomorphism from a group G
to a group and H be a subgroup of G. Then
(1) If |H| = n, then |φ(H)| divides n.
(2) If , then .
(3) If , then .
(4) If φ is onto and Ker φ = {e}, then φ is an isomorphism from G
to .