Isomorphism of Groups with Example

In this class, we discuss the Isomorphism of Groups with Examples.

The reader should have prior knowledge of the homomorphism of groups. Click Here.

Isomorphism:

A mapping f from a group [G *] to a group [H Δ] is said to be an isomorphism.

1) f should satisfy homomorphism. f(a * b) = f(a) Δ f(b)

2) f should satisfy one-to-one mapping

3) f should satisfy onto mapping.

Example:

G = [{1, w, w^2}, *]

H = [{0, 1, 2}, +mod3]

f: G -> H

f(1) = 0

f(w) = 1

f(w^2) = 2

The diagrams below show the operation tables.

f(a * b) = f(a) +mod3 f(b)

f(1 * w) = f(1) +mod3 f(w)

1 = 1

The above function satisfies one-to-one and onto. So, f is isomorphism.