I am following a note on complex reflections. There the notion of pseudoreflection groups are introduced. A pseuforeflection $\sigma : \mathbb C^n \longrightarrow \mathbb C^n$ is a linear isomorphism such that the order of $\sigma$ is finite as an element of $GL_n (\mathbb C)$ and $\sigma$ fixes a hyperplane i.e. the rank of the matrix $(\text {Id} - \sigma)$ is $1.$
Examples include symmetric groups, dihedral groups, finite cyclic groups etc. Although I have understood that both the symmetric groups and the dihedral groups, being generated by transpositions, are pseudoreflection groups as transpositions are pseudoreflections I couldn't able to understand why finite cyclic groups are pseudoreflection groups as well. Any cyclic group can be thought of as a group of rotational symmetries of a regular $n$-gon. So generating sets of a cyclic group are singletons generated by rotations by an angle $\frac {2 \pi r} {n}$ where $0 \lt r \lt n$ with $\text {gcd} (r,n) = 1.$ Each of which are of finite order but does they ever fix a hyperplane? If we denote the vertices of a regular $n$-gon by $1,2, \cdots, n$ then any such generator can be completely determined by its action on the $n$-element set $\{1, 2, \cdots, n \}.$ So they are elements of $S_n$ and since each one them has order $n$ it follows that they precisely correspond to $n$-cycles in $S_n.$ The matrix representation of any such cycle is a permutation matrix $P$ which is obtained by permuting the columns of the $n \times n$ identity matrix according to the $n$-cycle and it's not true that $(Id - P)$ has rank $1$ unless $n = 2.$ Am I missing something? Any suggestion in this regard would be warmly appreciated.
Thanks in advance.