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 Γ = ( G , S ) \Gamma=(G,S) , where 𝐺 is a group and S = { ρ 0 , … , ρ r − 1 } S=\{\rho_{0},\ldots,\rho_{r-1}\} is an ordered set of involutions generating 𝐺 and satisfying the following commuting property: for all i , j ∈ { 0 , … , r − 1 } i,j\in\{0,\ldots,r-1\} , | i − j | ≠ 1 \lvert i-j\rvert\neq 1 implies ( ρ i ⁢ ρ j ) 2 = 1 (\rho_{i}\rho_{j})^{2}=1 . When 𝑆 is an independent set, the rank of Γ 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 ≡ 0 , 1 , 4 ⁢ ( mod ⁢ 5 ) n\equiv 0,1,4\ (\mathrm{mod}\ {5}) .