# Fixed Points of Automorphisms of Certain Non-Cyclic p-Groups and the Dihedral Group

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

**Conjecture**

**1.**

## 2. Preliminaries

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

**Remark**

**1.**

**Definition**

**6.**

**Definition**

**7.**

**Definition**

**8.**

**Definition**

**9.**

**Definition**

**10.**

## 3. Proof of Conjecture 1

**Theorem**

**1.**

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

**Lemma**

**3.**

**Proof.**

**Remark**

**2.**

**Proposition**

**1.**

**Proof.**

**Lemma**

**4.**

**Proof.**

**Lemma**

**5.**

**Proof.**

**Proposition**

**2.**

**Proof.**

**Proposition**

**3.**

**Proof.**

**Theorem**

**2.**

**Remark**

**3.**

## 4. Number of Fixed-Point-Free Automorphisms of ${\mathbf{Z}}_{{\mathit{p}}^{\mathit{a}}}\oplus {\mathbf{Z}}_{{\mathit{p}}^{\mathit{b}}}$

**Proposition**

**4.**

**Proof.**

**Remark**

**4.**

## 5. $\mathit{\theta}$ Values for the Dihedral Group ${\mathit{D}}_{\mathbf{2}\mathit{p}}$

**Remark**

**5.**

**Example**

**1.**

**Theorem**

**3.**

**Proof.**

**Example**

**2.**

**Remark**

**6.**

**Question**. Let p be a prime number and ${a}_{1},\dots ,{a}_{n}$ be distinct positive integers. Is there an exact formula for $\theta \left({\displaystyle \underset{i=1}{\overset{n}{\u2a01}}{\mathbf{Z}}_{{p}^{{a}_{i}}},d}\right)$?

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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g | ${\mathit{f}}_{1,0}\left(\mathit{g}\right)$ | ${\mathit{f}}_{1,1}\left(\mathit{g}\right)$ | ${\mathit{f}}_{1,2}\left(\mathit{g}\right)$ | ${\mathit{f}}_{1,3}\left(\mathit{g}\right)$ | ${\mathit{f}}_{3,0}\left(\mathit{g}\right)$ | ${\mathit{f}}_{3,1}\left(\mathit{g}\right)$ | ${\mathit{f}}_{3,2}\left(\mathit{g}\right)$ | ${\mathit{f}}_{3,3}\left(\mathit{g}\right)$ |
---|---|---|---|---|---|---|---|---|

1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

a | a | a | a | a | ${a}^{3}$ | ${a}^{3}$ | ${a}^{3}$ | ${a}^{3}$ |

${a}^{2}$ | ${a}^{2}$ | ${a}^{2}$ | ${a}^{2}$ | ${a}^{2}$ | ${a}^{2}$ | ${a}^{2}$ | ${a}^{2}$ | ${a}^{2}$ |

${a}^{3}$ | ${a}^{3}$ | ${a}^{3}$ | ${a}^{3}$ | ${a}^{3}$ | a | a | a | a |

b | b | $ab$ | ${a}^{2}b$ | ${a}^{3}b$ | b | $ab$ | ${a}^{2}b$ | ${a}^{3}b$ |

$ab$ | $ab$ | ${a}^{2}b$ | ${a}^{3}b$ | b | ${a}^{3}b$ | b | $ab$ | ${a}^{2}b$ |

${a}^{2}b$ | ${a}^{2}b$ | ${a}^{3}b$ | b | $ab$ | ${a}^{2}b$ | ${a}^{3}b$ | b | $ab$ |

${a}^{3}b$ | ${a}^{3}b$ | b | $ab$ | ${a}^{2}b$ | $ab$ | ${a}^{2}b$ | ${a}^{3}b$ | b |

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## Share and Cite

**MDPI and ACS Style**

Hayat, U.; López-Aguayo, D.; Abbas, A.
Fixed Points of Automorphisms of Certain Non-Cyclic *p*-Groups and the Dihedral Group. *Symmetry* **2018**, *10*, 238.
https://doi.org/10.3390/sym10070238

**AMA Style**

Hayat U, López-Aguayo D, Abbas A.
Fixed Points of Automorphisms of Certain Non-Cyclic *p*-Groups and the Dihedral Group. *Symmetry*. 2018; 10(7):238.
https://doi.org/10.3390/sym10070238

**Chicago/Turabian Style**

Hayat, Umar, Daniel López-Aguayo, and Akhtar Abbas.
2018. "Fixed Points of Automorphisms of Certain Non-Cyclic *p*-Groups and the Dihedral Group" *Symmetry* 10, no. 7: 238.
https://doi.org/10.3390/sym10070238