Isomorphism

A function $\theta :G\to H$ between groups $(G,\ast )$ and $(H,\circ )$ is called an isomorphism if:

• $\theta$ is a homomorphism, i.e. $\theta (g_1\ast g_2)=\theta (g_1)\circ \theta (g_2)$ for all $g_1,g_2\in G$,
• $\theta$ is bijective (one-to-one and onto).

In this case we say $G$ and $H$ are isomorphic and write $G\cong H$.