Definition of symmetry factor $p$ in Feynmans diagrams symmetry factor in Coleman's "Introduction to Many-Body Physics"

Question

I'm trying to digest Coleman's 7.2.1 chapter about symmetry factors. Everything is clear up to point 4 where he introduces symmetry factor $p$ as the "dimension of the group of permutations under which the connectivity of the diagram is unchanged". It is not clear at all to me what he means here. Firstly, permutations of what we are talking about? Secondly, the examples he provides seem to contradict this definition.

In equation 7.63 he gives two permutations in cycle notation, but they are the same transpositions! So shouldn't $p$ be equal to 1?

Same in 7.65, two equivalent cycles. And in 7.67 he gives 4 permutations, but again the first and the second cycles are equivalent as well as the third and the fourth. So shouldn't $p$ be equal here to 2?

Could you maybe provide an alternative definition of $p$ that could be anyhow understandable and maybe apply it to these examples?

Answer 1

1. For a Feynman diagram with $n$ vertices, Coleman's notation for a permutation $\pi\in S_n$ is the image $(\pi(1) \pi(2)\ldots \pi(n))$; the notation should not be read as cycle notation for a cyclic permutation.

2. The symmetry factor $p=|{\cal G}|$ is the order of the invariant subgroup ${\cal G}\subseteq S_n$.