Rewritable groups

A group G is said to have the n-rewritable property Qn if for all elements g1,g2,…,gn∈G, there exist two distinct permutations σ,τ∈Symn such that gσ(1)gσ(2)⋯gσ(n)=gτ(1)gτ(2)⋯gτ(n). We show here that if G satisfies Qn, then G has a characteristic subgroup N such that |G:N| and |N′| are both finite and have sizes bounded by functions of n. This extends the result of Blyth (1988) in [3] which asserts that if G satisfies Qn and if Δ is the finite conjugate center of the group, then |G:Δ| and |Δ′| are both finite with |G:Δ| bounded by a function of n. As a consequence, any group with Qn satisfies the permutational property Pm with m bounded by a function of n.