What is `PermutationGroup` in graph automorphism group Mathematica output?

Question

PermutationGroup[{
    Cycles[{{3,6},{5,7},{8,10}}],
    Cycles[{{2,5,10,9,8,7},{3,6,4}}],
    Cycles[{{1,2},{3,7},{5,6}}]
}]

I used the Mathematica function GraphAutomorphismGroup to identify the Petersen graph's automorphism group, and it returned the result above. I have not taken group theory. Could someone explain to me how to interpret those cycles in a layman's terms?

Answer 1

I will give only a brief explanation here, but I'll provide references to help the reader.

The permutation group $S_n$ is the group of all ways that the symbols $\{1, \dotsc, n\}$ can be permuted. Since the Petersen graph has ten vertices, it's automorphism group will be a subgroup of $S_{10}$, where the symbols correspond to the vertices. Some elements of $S_n$ are called cycles because they permute the elements in a cyclic fashion. So the permutation that sends 3 to 6, 6 to 4, and 4 to 3 is a cycle, and can be written concisely as $(3\,6\,4)$. Now we have a fun fact that every element of $S_n$ can be written as a product of disjoint cycles, so a typical element of $S_n$ might look like $$(2\,5\,10\;9\,8\,7)(3\,6\,4)\,.$$ Like I said before, the automorphism group of the Petersen graph is a subgroup of $S_{10}$. When Mathematica says that it's the subgroup

PermutationGroup[{
    Cycles[{{3,6},{5,7},{8,10}}],
    Cycles[{{2,5,10,9,8,7},{3,6,4}}],
    Cycles[{{1,2},{3,7},{5,6}}]
}]

Mathematica means that it's the smallest subgroup of $S_{10}$ containing the three permutations: $$(3\,6)(5\,7)(8\,10)\qquad(2\,5\,10\;9\,8\,7)(3\,6\,4)\qquad(1\,2)(3\,7)(5\,6)$$