---
tag: 268FWM8ZMJ
kind: proof
name: Proof of Frucht's theorem
proves: W7Q3ZPF46B
sources:
  - wikipedia: en:26259599@r1327421226
    locator: Frucht's theorem, section Proof idea
    note: adopted
    mention: wikipedia://en/26259599#XOGT
    sections: [Proof]
---

## Proof

The main idea of the proof is to observe that the [[Cayley graph|0GYQK9NDJF]] of *G*, with the addition of colors and orientations on its edges to distinguish the generators of *G* from each other, has the desired automorphism group. Therefore, if each of these edges is replaced by an appropriate subgraph, such that each replacement subgraph is itself asymmetric and two replacements are isomorphic if and only if they replace edges of the same color, then the undirected graph created by performing these replacements will also have *G* as its symmetry group.