Investigating transversals as generating sets for groups

Abstract

In [3] is was shown that for any group G whose rank (i.e., minimal number of generators) is at most 3, and any finite index subgroup H G with index [G : H] rank(G), one can always find a left-right transversal of H which generates G. In this paper we extend this result to groups of rank at most 4. We also extend this to groups G of arbitrary (finite) rank r provided all the non-trivial divisors of [G : CoreG(H)] are at least 2r − 1. Finally, we extend this to groups G of arbitrary (finite) rank provided H is malnormal in G.