Group isomorphism

Definition

Let $G$ and $H$ be groups. A group isomorphism from $G$ to $H$ is a group homomorphism $\varphi \colon G \to H$ that is a bijection. The groups $G$ and $H$ are isomorphic, written $G \cong H$, when there is an isomorphism from $G$ to $H$.

Remarks

The inverse map of an isomorphism is again a homomorphism, thus an isomorphism. Isomorphism is an equivalence relation on groups. The exponential map is an isomorphism from $(\mathbb{R}, +)$ to the positive real numbers with multiplication. Isomorphic groups have the same order, and corresponding elements have the same order.

Explanations

L1 human

Two groups are isomorphic when they are the same group with different names for the elements. An isomorphism is a dictionary that translates the names and keeps all products correct. For example, the rotations of a triangle and the integers modulo $3$ under addition are isomorphic.