For a given positive integer $n$, the structure, i.e. the number of cycles of various lengths, as well as possible chains, of the automorphisms of the groups $(\Z^n, +)$ and $(\Z_p^n,+)$, \ $p$ prime, is studied. In other words, necessary and sufficient conditions on a bijection $f : A \ra A$, where $ A $ is countably infinite (alternatively, of order $p^n$), are determined so that $A$ can be endowed with a binary operation $*$ such that $(A,*)$ is a group isomorphic to $(\Z^n,+)$ (alternatively, $(\Z_p^n,+)$) and such that $f\in \Aut(A)$.
Automorphism, abelian group
KLERK, BEN-EBEN DE and MEYER, JOHAN H.
"When is a permutation of the set $\Z^n$ (resp. $\Z_p^n$, $p$ prime) an automorphism of the group $\Z^n$ (resp. $\Z_p^n$)?,"
Turkish Journal of Mathematics: Vol. 42:
6, Article 11.
Available at: https://journals.tubitak.gov.tr/math/vol42/iss6/11