Permutation representations of finite abelian groups What is a good source to study from about permutation representations of finite abelian groups, specifically $\mathbb{Z}_{p}$? If reference for the specific topic is not available, I would like to study about permutation representations in general (hopefully from a combinatorial viewpoint).
 A: Permutation representations of finite cyclic groups are pretty simple.
If $G$ is a cyclic group of order $d$, and $g$ is a generator of $G$, then a permutation representation in $S_n$ is determined by the image of $g$, which must be a permutation of order dividing $d$.  A permutation of order dividing $d$ is one whose cycle decomposition consists of cycles whose orders divide $d$.  And a permutation is determined up to conjugation by its cycle type.
So, for instance, the permutation representations of a cyclic group of order 6 in $S_5$ would be (up to isomorphism)


*

*$\mathbb{1}$

*$(\cdot\cdot)$

*$(\cdot\cdot)(\cdot\cdot)$

*$(\cdot\cdot\cdot)$

*$(\cdot\cdot\cdot)(\cdot\cdot)$


When $d$ is prime, things are even simpler, because the cycles can only be of order $d$.  For instance, the permutation representations of a cyclic group of order 5 in $S_{20}$ are


*

*$\mathbb{1}$

*$(\cdot\cdot\cdot\cdot\cdot)$

*$(\cdot\cdot\cdot\cdot\cdot)(\cdot\cdot\cdot\cdot\cdot)$

*$(\cdot\cdot\cdot\cdot\cdot)(\cdot\cdot\cdot\cdot\cdot)(\cdot\cdot\cdot\cdot\cdot)$

*$(\cdot\cdot\cdot\cdot\cdot)(\cdot\cdot\cdot\cdot\cdot)(\cdot\cdot\cdot\cdot\cdot)(\cdot\cdot\cdot\cdot\cdot)$


And the irreducible representations are yet simpler: they just correspond to a single $m$-cycle, where $m$ divides $d$.
