The maximal rank of a string group generated by involutions for alternating groups

TL;DR

This paper establishes an upper bound on the rank of string groups generated by involutions over alternating groups, with the bound being tight for specific congruence classes of n.

Contribution

It provides the first explicit upper bound for the rank of such groups over alternating groups, identifying cases where the bound is tight.

Findings

Derived an explicit upper bound for the rank of SGGI over alternating groups.

Proved the bound is tight when n ≡ 0,1,4 mod 5.

Enhanced understanding of involution-generated groups in symmetric and alternating groups.

Abstract

A string group generated by involutions, or SGGI, is a pair $\Gamma=(G, S)$, where $G$ is a group and $S=\{\rho_0,\ldots, \rho_{r-1}\}$ is an ordered set of involutions generating $G$ and satisfying the commuting property: $$\forall i,j\in\{0,\ldots, r-1\}, \;|i-j|\ne 1\Rightarrow (\rho_i\rho_j)^2=1.$$ When $S$ is an independent set, the rank of $\Gamma$ is the cardinality of $S$. We determine an upper bound for the rank of an SGGI over the alternating group of degree $n$. Our bound is tight when $n\equiv 0,1,4\pmod 5$.