2025 : 4 : 20
Azizolah Azad

Azizolah Azad

Academic rank: Associate Professor
ORCID: https://orcid.org/0000-0002-7950-0977
Education: PhD.
ScopusId: 14420904600
HIndex: 0/00
Faculty: Science
Address: Arak University
Phone:

Research

Title
Diameter of a direct power of alternating groups
Type
JournalPaper
Keywords
Diameter of a group, rank of a group, non-abelian simple groups
Year
2025
Journal Bol. Soc. Paran. Mat
DOI
Researchers Azizolah Azad ، Nasim Karimi

Abstract

In this paper we estimate the diameter of a direct power of alternating groups Ak for k ≥ 4. We show that there exists a generating set of minimum size for An 4 , for which the diameter of An 4 is O(n). For k ≥ 5, we show that there exists a generating set of minimum size for A2 k , for which the diameter of A2 k is at most O(ke(c+1)(log k) 4 log log k) , for an absolute constant c > 0. Finally for 1 ≤ n ≤ 8, we provide generating sets of size two for An 5 and we show that the diameter of An 5 with respect to those generating sets is O(n). These results leads us to the sense that the best upper bound known for the diameter of the direct power of non-abelian simple groups (specially alternating groups), i.e. O(n 3 ) [5], may be improved to O(n). Furthermore, these results are more pieces of evidence for a conjecture which has been presented in [9] in 2015.