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, and that the 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 ⋅ 2n−1 , and h
contains no primitive elements of g, then h is normal in g and g/h ≅ c2n .