Transversals as Generating Sets in Finitely Generated Groups

Abstract

We explore transversals of finite index subgroups of finitely generated groups. We show that when H is a subgroup of a rank n group G and H has index at least n in G then we can construct a left transversal for H which contains a generating set of size n for G; this construction is algorithmic when G is finitely presented. We also show that, in the case where G has rank n ≤ 3, there is a simultaneous left-right transversal for H which contains a generating set of size n for G. We finish by showing that if H is a subgroup of a rank n group G with index less than 3⋅2^n-1, and H contains no primitive elements of G, then H is normal in G and G/H ≅ Cn2.