# Antiautomorphisms and Biantiautomorphisms of Some Finite Abelian Groups

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## Abstract

**:**

## 1. Introduction

**2014**, proposed by Gaitanas Konstantinos in [1]:

**Problem 2014**. For every integer $n\ge 2$, let $({\mathbf{Z}}_{n},+)$ be the additive group of integers modulo n. Define an antimorphism of ${\mathbf{Z}}_{n}$ to be any function $f:{\mathbf{Z}}_{n}\to {\mathbf{Z}}_{n}$ such that $f\left(x\right)-f\left(y\right)\ne x-y$ whenever $x,y$ are distinct elements of ${\mathbf{Z}}_{n}$. We say that f is an antiautomorphism of ${\mathbf{Z}}_{n}$ if f is a bijective antimorphism of ${\mathbf{Z}}_{n}$. For what values of n does ${\mathbf{Z}}_{n}$ admit an antiautomorphism?

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

**Proof.**

**Proposition**

**3.**

**Proof.**

**Remark**

**1.**

**Proposition**

**4.**

**Proof.**

**Remark**

**2.**

## 2. Antiautomorphisms

**Definition**

**1**

**.**Let G be an abelian group and let $f:G\to G$ be any function. We say that f is an antimorphism if the map $i{d}_{G}-f$ is injective. We say that an antimorphism f is an antiautomorphism of G if f is a bijection.

**Remark**

**3.**

**Proposition**

**5.**

**Proposition**

**6.**

**Proof.**

**Lemma**

**1.**

**Proposition**

**7.**

**Proof.**

**Proposition**

**8.**

**Proof.**

**Remark**

**4.**

**Proposition**

**9.**

**Proof.**

## 3. Biantiautomorphisms

**Definition**

**2**

**.**Let G be a finite abelian group and let f be an antiautomorphism of G. We say that f is a biantiautomorphism of G if f is also a linear map.

**Proposition**

**10.**

**Proof.**

**Remark**

**5.**

**Lemma**

**2.**

**Proof.**

**Proposition**

**11.**

**Proof.**

**Theorem**

**1.**

**Proof.**

**Definition**

**3**

**.**Let f be an automorphism of a finite abelian group G. We say that f is a fixed point free automorphism of G if $f\left(x\right)\ne x$ for all $x\in G\backslash \left\{0\right\}$.

**Proposition**

**12.**

**Proof.**

**Lemma**

**3.**

**Lemma**

**4.**

**Proof.**

**Theorem**

**2.**

- 1.
- If G has no elements of order 2, then G admits an antiautomorphism.
- 2.
- If G has exactly one element of order 2, then G does not admit antiautomorphisms.
- 3.
- If $G={\mathbf{Z}}_{{2}^{m}}^{n}$, where $n\ge 2$ and $m\ge 1$, then G admits an antiautomorphism.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Problem 2014. Math. Mag.
**2017**, 90, 75–76. - Górowski, J.; Lomnicki, A. Simple proofs of some generalizations of the Wilson’s theorem. Annales Universitatis Paedagogicae Cracoviensis
**2014**, 13, 7–14. [Google Scholar] [CrossRef] - Izumi, M.; Kosaki, H. Kac Algebras Arising from Composition of Subfactors: General Theory and Classification. In Memoirs of the American Mathematical Society; American Mathematical Society: New York, NY, USA, 2002; Volume 158. [Google Scholar]
- Mayr, P. Finite fixed point free automorphism groups. Master’s Thesis, University Linz, Linz, Austria, 1999. [Google Scholar]
- Ryan, C. Linear sections of the general linear group: A geometric approach. Discret. Appl. Math.
**1992**, 35, 81–86. [Google Scholar] [CrossRef] [Green Version]

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## Share and Cite

**MDPI and ACS Style**

López-Aguayo, D.; López Aguayo, S.
Antiautomorphisms and Biantiautomorphisms of Some Finite Abelian Groups. *Symmetry* **2020**, *12*, 294.
https://doi.org/10.3390/sym12020294

**AMA Style**

López-Aguayo D, López Aguayo S.
Antiautomorphisms and Biantiautomorphisms of Some Finite Abelian Groups. *Symmetry*. 2020; 12(2):294.
https://doi.org/10.3390/sym12020294

**Chicago/Turabian Style**

López-Aguayo, Daniel, and Servando López Aguayo.
2020. "Antiautomorphisms and Biantiautomorphisms of Some Finite Abelian Groups" *Symmetry* 12, no. 2: 294.
https://doi.org/10.3390/sym12020294