#
A Computational Approach to Verbal Width for Engel Words in Alternating Groups^{ †}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Engel Chains

**Definition**

**1.**

**Lemma**

**1.**

**Proof.**

**Definition**

**2.**

**Lemma**

**2.**

**Proof.**

**Lemma**

**3.**

- 1.
- For every $z\in {C}_{{A}_{n}}\left(y\right)\delta $ we have that $[\delta ,y]=[z,y]$.
- 2.
- For every $z\in \delta {C}_{{A}_{n}}\left(y\right)$ we have that $[z,y]\in {B}^{y}\left(x\right)$.

**Proof.**

**Lemma**

**4.**

**Proof.**

**Definition**

**3.**

**Lemma**

**5.**

**Proof.**

**Lemma**

**6.**

**Proof.**

- $Z\subset {N}_{1}\subset {N}_{2}\subset {N}_{3}\subset \dots $
- $Z\subset {N}_{{A}_{p}}\left(Z\right)={N}_{1}\subset {N}_{{A}_{p}}\left({N}_{1}\right)=\tilde{{N}_{2}}\subset {N}_{{A}_{p}}\left({N}_{2}\right)=\tilde{{N}_{3}}\subset \dots $

**Lemma**

**7.**

**Proof.**

**Corollary**

**1.**

**Proof.**

## 3. Engel Graphs

- Given ${z}_{1},{z}_{2}\in {V}_{n}^{y}$, there exists an arrow from ${z}_{1}$ to ${z}_{2}$ if an only if ${C}_{{A}_{n}}\left(y\right)[{z}_{1},y]={C}_{{A}_{n}}\left(y\right){z}_{2}$.

**Definition**

**4.**

- If we consider a path of length k in the graph, starting in the node ${C}_{{A}_{n}}\left(y\right){z}_{1}$ and finishing in the node ${C}_{{A}_{n}}\left(y\right){z}_{k+1}$, we have that ${E}_{k}({z}_{1},y)=[{z}_{k+1},y]$. Once the graph is built, it is possible to easily compute Engel words of high lengths.
- Reciprocally, if we want to compute ${E}_{k}(x,y)$, it is enough to consider a path of length k starting in the node ${C}_{{A}_{n}}\left(y\right)x$ and commute by y any element of the coset associated to the last node of the path ${C}_{{A}_{n}}\left(y\right){z}_{{k}_{1}}$. We have that$${E}_{k}(x,y)=[{z}_{k-1},y].$$
- We can study the ’dynamic’ of the set ${\left\{{E}_{m}(\xb7,y)\right\}}_{m\ge 0}$ by studying the ’dynamic’ of the graph $({V}_{n}^{y},\mathbb{A})$.

**Lemma**

**8.**

**Proof.**

**Corollary**

**2.**

**Proof.**

**Lemma**

**9.**

**Proof.**

**Corollary**

**3.**

**Lemma**

**10.**

**Proof.**

## 4. Engel Graphs for Small Alternating Groups

**Lemma**

**11.**

**Theorem**

**1.**

- Using the algorithm described above in GAP for ${A}_{6}$ and $y=(1,2,3,4,5)$, we get that the types of permutations in ${A}_{6}$ which do not appear in $\mathsf{\Omega}$ are$$\left\{\right(1,2\left)\right(3,4),(1,2,3),(1,2,3\left)\right(4,5,6\left)\right\}.$$Applying Lemma 11, we can get Theorem 1 for the group ${A}_{6}$.
- If we take ${A}_{n}$, with $7\le n\le 14$ and we repeat the same process for $y=(1,2,3,\dots ,n)$, if n is odd but $y=(1,2,3,\dots ,n-1)$ if n is even, there is only one type of permutation that does not appear in the set $\mathsf{\Omega}$: $\left\{\right(1,2\left)\right(3,4\left)\right\}$.And again, we can easily get the Theorem 1 for the groups ${A}_{n}$, with $7\le n\le 14$.

**Theorem**

**2.**

## Funding

## Conflicts of Interest

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Alternating Group | Max. Length |
---|---|

${A}_{5}$ | 2 |

${A}_{6}$ | 2 |

${A}_{7}$ | 2 |

${A}_{8}$ | 2 |

${A}_{9}$ | 3 |

${A}_{10}$ | 3 |

${A}_{11}$ | 2 |

${A}_{12}$ | 2 |

${A}_{13}$ | 2 |

${A}_{14}$ | 2 |

Group | Conj. Cl. Not Found | Run Time |
---|---|---|

${A}_{5}$ | $\{(1,2){(3,4)}^{{S}_{5}},{(1,2,3)}^{{S}_{5}}\}$ | 7 ms |

${A}_{6}$ | $\{(1,2){(3,4)}^{{S}_{6}},{(1,2,3)}^{{S}_{6}},$ | 18 ms |

$(1,2,3){(4,5,6)}^{{S}_{6}}\}$ | ||

${A}_{7}$ | $\{(1,2){(3,4)}^{{S}_{7}}\}$ | 40 ms |

${A}_{8}$ | $\{(1,2){(3,4)}^{{S}_{8}}\}$ | 201 ms |

${A}_{9}$ | $\{(1,2){(3,4)}^{{S}_{9}}\}$ | 4 s 12 ms |

${A}_{10}$ | $\{(1,2){(3,4)}^{{S}_{10}}\}$ | 40 s 809 ms |

${A}_{11}$ | $\{(1,2){(3,4)}^{{S}_{11}}\}$ | 5 min 37 s 139 m |

${A}_{12}$ | $\{(1,2){(3,4)}^{{S}_{12}}\}$ | 63 min 38 s 210 m |

${A}_{13}$ | $\{(1,2){(3,4)}^{{S}_{13}}\}$ | 21 h 6 min 54 s |

${A}_{14}$ | $\{(1,2){(3,4)}^{{S}_{14}}\}$ | approx. 12 days |

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**MDPI and ACS Style**

Martínez Carracedo, J.
A Computational Approach to Verbal Width for Engel Words in Alternating Groups. *Symmetry* **2019**, *11*, 877.
https://doi.org/10.3390/sym11070877

**AMA Style**

Martínez Carracedo J.
A Computational Approach to Verbal Width for Engel Words in Alternating Groups. *Symmetry*. 2019; 11(7):877.
https://doi.org/10.3390/sym11070877

**Chicago/Turabian Style**

Martínez Carracedo, Jorge.
2019. "A Computational Approach to Verbal Width for Engel Words in Alternating Groups" *Symmetry* 11, no. 7: 877.
https://doi.org/10.3390/sym11070877