---
tag: 085PND4KNT
kind: definition
name: Group isomorphism
aliases: [isomorphism-of-groups]
msc: [20A05]
defines: 08MG7H8JW5
uses: [0HV9Y7WG5R]
sources:
  - url: https://en.wikipedia.org/wiki/Group_isomorphism
    locator: Definition and notation
---

## Definition

Let $G$ and $H$ be [[groups|G5S1M9]].
A group isomorphism from $G$ to $H$ is a [[group homomorphism|0JYTG5K9Y4]] $\varphi \colon G \to H$ 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.