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

Abstract

A string group generated by involutions, or SGGI, is a pair $\Gamma =(G,S)$, where $G$ is a group and $S=\{{\rho }_0,\dots ,{\rho }_{r-1}\}$ is an ordered set of involutions generating $G$ and satisfying the commuting property: $$\forall i,j\in \{0,\dots ,r-1\},|i-j|\ne 1\Rightarrow ({\rho }_i{\rho }_j{)}^2=1.$$ 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 $n\equiv 0,1,4(\mathrm{mod}5)$.

28 pages