Diameter of a direct power of alternating groups

TL;DR

This paper investigates the diameter of direct powers of alternating groups, providing bounds and constructions for specific cases, and contributes evidence towards a broader conjecture on group diameters.

Contribution

It offers new bounds on the diameter of $A_n^k$ for various $n$ and $k$, including explicit generating sets and asymptotic estimates, advancing understanding of diameters in non-abelian simple groups.

Findings

Existence of generating sets of size two for $A_5^n$ with $O(n)$ diameter.

Bound of $O(n)$ for $A_4^n$ diameter with minimal generating sets.

Upper bound of $O(ne^{(c+1)( ext{log} )^4 ext{log} ext{log} n})$ for $A_n^2$, $n extgreater 4$.

Abstract

So far, it has been proven that if $G$ is an abelian group , then the diameter of $G^n$ with respect to any generating set is $O(n)$; and if $G$ is nilpotent, symmetric or dihedral, then there exists a generating set of minimum size, for which the diameter of $G^n$ is $O(n)$ \cite{Karimi:2017}. In \cite{Dona:2022} it has been proven that if $G$ is a non-abelian simple group, then the diameter of $G^n$ with respect to any generating set is $O(n^3)$. In this paper we estimate the diameter of direct power of alternating groups $A_n$ for $n \geq 4$, i.e. a class of non-abelian simple groups. We show that there exist a generating set of minimum size for $A_4^n$, for which the diameter of $A_4^n$ is $O(n)$. For $n \geq 5$, we show that there exists a generating set of minimum size for $A_n^2$, for which the diameter of $A_n^2$ is at most $O(ne^{(c+1) (\log \,n)^4 \log \log n})$ , for an absolute constant $c >0$. Finally for $ 1\leq n \leq 8 $, we provide generating sets of size two for $A_5^n$ and we show that the diameter of $A_5^n$ with respect to those generating sets is $O(n)$. These results are more pieces of evidence for a conjecture which has been presented in \cite{Karimithesis:2015} in 2015.