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 $H\le G$ with index $[G:H]\ge rank(G)$, 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 $[G:cor{e}_G(H)]$ are at least $2r-1$. Finally, we extend this to groups $G$ of arbitrary (finite) rank provided $H$ is malnormal in $G$.

36 pages. This is the first version, comments and suggestions are welcome