# Wolfram Function Repository

Instant-use add-on functions for the Wolfram Language

Function Repository Resource:

Find an undirected graph whose automorphism group is isomorphic to the given group

Contributed by:
Daniel McDonald

ResourceFunction["AutomorphismGraph"][ finds an undirected graph whose automorphism group is isomorphic to the group |

The Cayley graph of a group G with respect to some generating set S is a directed graph whose vertices are the elements of G, with an edge of color c_{s} directed from g to gs for each group element g∈G and generator s∈S. If C is the Cayley graph of G with respect to S, then G is isomorphic to the subgroup of the automorphism group of C consisting of the color-preserving automorphisms. An undirected graph H whose automorphism group is G can be constructed from C by replacing each edge of color c_{s} with an appropriate asymmetric graph H_{s}. Hence every finite group is isomorphic to the automorphism group of some undirected graph; this is the statement of Frucht’s theorem.

The group is typically a PermutationGroup or similar group like a SymmetricGroup, AlternatingGroup, CyclicGroup or DihedralGroup.

Specify a group:

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Construct the graph provided whose automorphism group is isomorphic to our group:

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Find the graph's automorphism group:

In[3]:= |

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Verify that an isomorphism exists between the two groups:

In[4]:= |

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Wolfram Language 13.0 (December 2021) or above

- 1.0.0 – 29 November 2023

This work is licensed under a Creative Commons Attribution 4.0 International License