Suppose
that φ is an isomorphism from a group G
onto another group 
. Then
(1) G is Abelian if and only if is Abelian.
(2) G is cyclic if and only if is cyclic.
(3)
φ-1 is an isomorphism from onto G.
(4)
If K ≤ G, then 
≤
.
Suppose
that φ is an isomorphism from a group G
onto another group . Then
(1) G is Abelian if and only if is Abelian.
(2) G is cyclic if and only if is cyclic.
(3)
φ-1 is an isomorphism from onto G.
(4)
If K ≤ G, then ≤
.