A mapping f from a group (G, o) to a group (G', &lowast;) is group h...

f(a &lowast; b) = f(a) o f(b) ∀ a, b ∈ G

f(a o b) = f(a) &lowast; f(b) ∀ a, b ∈ G

f -1 (a &lowast; b) = f -1 (a) o f -1 (b) ∀ a, b ∈ G

f(a o b) = f(a) &lowast; f -1 (b) ∀ a, b ∈ G

A mapping f from a group (G, o) to a group (G', ∗) is group homomorphism from G to G' if

f(a ∗ b) = f(a) o f(b) ∀ a, b ∈ G

f(a o b) = f(a) ∗ f(b) ∀ a, b ∈ G

f-1(a ∗ b) = f-1(a) o f-1(b) ∀ a, b ∈ G

f(a o b) = f(a) ∗ f-1(b) ∀ a, b ∈ G