# When is a permutation of the set $\Z^n$ (resp. $\Z_p^n$, $p$ prime) an automorphism of the group $\Z^n$ (resp. $\Z_p^n$)?

Öz 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 $f\in \Aut(A)$.
Anahtar Kelimeler:

## Automorphism, abelian group

#### Kaynak Göster

 Bibtex @ { tbtkmath575207, journal = {Turkish Journal of Mathematics}, issn = {1300-0098}, eissn = {1303-6149}, address = {}, publisher = {TÜBİTAK}, year = {2018}, volume = {42}, pages = {2965 - 2978}, doi = {}, title = {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\$)?}, key = {cite}, author = {Klerk, Ben-eben De and Meyer, Johan H.} } APA Klerk, B , Meyer, J . (2018). 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 , 42 (6) , 2965-2978 . MLA Klerk, B , Meyer, J . "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 42 (2018 ): 2965-2978 < Chicago Klerk, B , Meyer, J . "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 42 (2018 ): 2965-2978 RIS TY - JOUR T1 - When is a permutation of the set $\Z^n$ (resp. $\Z_p^n$, $p$ prime) an automorphism of the group $\Z^n$ (resp. $\Z_p^n$)? AU - Ben-eben De Klerk , Johan H. Meyer Y1 - 2018 PY - 2018 N1 - DO - T2 - Turkish Journal of Mathematics JF - Journal JO - JOR SP - 2965 EP - 2978 VL - 42 IS - 6 SN - 1300-0098-1303-6149 M3 - UR - Y2 - 2021 ER - EndNote %0 Turkish Journal of Mathematics When is a permutation of the set $\Z^n$ (resp. $\Z_p^n$, $p$ prime) an automorphism of the group $\Z^n$ (resp. $\Z_p^n$)? %A Ben-eben De Klerk , Johan H. Meyer %T When is a permutation of the set $\Z^n$ (resp. $\Z_p^n$, $p$ prime) an automorphism of the group $\Z^n$ (resp. $\Z_p^n$)? %D 2018 %J Turkish Journal of Mathematics %P 1300-0098-1303-6149 %V 42 %N 6 %R %U ISNAD Klerk, Ben-eben De , 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 42 / 6 (Aralık 2018): 2965-2978 . AMA Klerk B , Meyer J . 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. 2018; 42(6): 2965-2978. Vancouver Klerk B , Meyer J . 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. 2018; 42(6): 2965-2978. IEEE B. Klerk ve J. Meyer , "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, c. 42, sayı. 6, ss. 2965-2978, Ara. 2018