looking for examples for actions of the cyclic group $Z/m$ on abelian groups

Question

Do you have any examples for actions of the cyclic group $Z/m$ on abelian groups? Especially, examples, where the action respects the group structure of the abelian group A, i.e $g.(a+b) = g.a + g.b , g \in \mathbb{Z} / m , \ a,b \in A$, would be interesting.

Answer 1

A group action of $Z/m$ on an Abelian group $A$ that respects the group structure of $A$ can be formed from a homomorphism from $Z/m$ to Aut$(A)$.

For a given $m$, we can take $A$ to be the direct sum of $m$ copies of some cyclic group such as $Z$. Let the generators of $A$ be $a_1, a_2,...,a_m$ and let the elements of $Z/m$ be $0, 1, ... ,m-1$ then the group action is $(k, a_j)$ goes to $a_{k+j \mod m} $