Group Isomorphism

Let $$G$$ and $$G’$$ be any two groups with binary operation $$ \circ $$ and $$ \circ ‘$$, respectively. If there exists a one-one onto mapping $$f:G \to G’$$ such that
\[f\left( {a \circ b} \right) = f\left( a \right) \circ ‘f\left( b \right),\,\,\,\forall a,b \in G\]

In this case, the group $$G$$ is said to be isomorphic to the group $$G’$$, and the mapping $$f$$ is said to be an isomorphism. If $$G$$ is isomorphic to $$G’$$, we write $$G \simeq G’$$ or $$G \cong G’$$.

In other words, a group $$G$$ is isomorphic to the group $$G’$$ if there exists a one-one onto mapping of $$G$$ and $$G’$$ such that the image of the product of two elements is the product of the images of the elements with respect to the composition in the respective group.

The last condition may also be stated as follows:

If$$ab = c$$ where $$a,b,c \in G$$ and $$f\left( a \right) = a’,\,\,\,f\left( b \right) = b’,\,\,\,f\left( c \right) = c’$$ then $$a’b’ = c’$$ where $$a’,b’,c’ \in G’$$.