Automorphism group of a cycle graph in Mathematica

Question

From this lecture note, it is known that the automorphism group of the $n$-cycle $C_n$ is the dihedral group $D_n$ with $2 n$ elements.

I use the following code to create a cycle graph of $6$ nodes.

g = CycleGraph[6]

It gives me the following output.

Now I use the following code to compute its automorphism group.

GraphAutomorphismGroup[g]

It gives me the following output.

PermutationGroup[{Cycles[{{2, 6}, {3, 5}}], Cycles[{{1, 2, 3, 4, 5, 6}}]}]

On the other hand if I directly create a dihedral group it looks as follows.

In[15]:= DihedralGroup[6] // GroupGenerators Out[15]= {Cycles[{{1, 6}, {2, 5}, {3, 4}}], Cycles[{{1, 2, 3, 4, 5, 6}}]}

So, why isn't the dihedral group same as the automorphism group of the cycle graph?

M.R. · Accepted Answer · 2016-09-26 04:11:09Z

You can test that the elements are the same:

g1 = GraphAutomorphismGroup[g]; g2 = DihedralGroup[6]; SameQ @@ (GroupElements /@ {g1, g2})

yode · Accepted Answer · 2022-07-02 14:59:19Z

GraphAutomorphismGroup[CycleGraph[6]] and DihedralGroup[6] are automorphisms. We can use Equal to check it:

GraphAutomorphismGroup[CycleGraph[6]]==DihedralGroup[6]

True

2 Answers 2