Group isomorphism

Let $G,H$ be groups. A group isomorphism is a group homomorphism $\varphi\colon G\to H$ that is bijective as a bijective function .

If $\varphi$ is an isomorphism, then its inverse function $\varphi^{-1}\colon H\to G$ is also a group homomorphism, so $\varphi$ gives a structure-preserving identification between $G$ and $H$. One writes $G\cong H$ to denote that there exists a group isomorphism between them.

Examples:

• The map $\mathbb{Z}\to 2\mathbb{Z}$ given by $n\mapsto 2n$ is a group isomorphism (additive groups).
• For each $n\ge 1$, any cyclic group of order $n$ is isomorphic to $\mathbb{Z}/n\mathbb{Z}$.
• The map $G\to G$ given by $g\mapsto xgx^{-1}$ (for fixed $x\in G$) is an isomorphism (an inner automorphism).