In the book of Algebraic Graph Theory by Godsil, at page 8, it is stated that
However, the group $\langle g\rangle$ is of order $n$ and the group $\langle h\rangle $ is of order 2, so shouldn't we have $$|Aut(C_n)| \geq n+2$$ ? I mean I couldn't understand the argument that the author makes. Can some help me to understand the reason why we have $$|Aut(C_n)| \geq 2 |R| = 2n$$ ?
In addition to the automorphisms $$ 1,g,g^2,\dots,g^{n-1} $$ there are also $$ h,hg,hg^2,\dots,hg^{n-1} $$ making for at least $2n$ automorphisms. The first list is $R$, the second list is the set $hR=\{hr:r\in R\}$. This is called a coset of R, and is the second coset that text mentions.