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, 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 C n 2 .2010 Mathematics subject classification: primary 20F05; secondary 20E99.

Investigating transversals as generating sets for groups

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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.

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“…It is also true that such a transversal always exists, as shown in [3,Theorem 3.7]. And clearly the reverse statement is true: if there exists a left transversal of H which generates G then rank(G) [G : H].…”

“…This question was first studied in [3], where it was shown in [3,Theorem 3.11] that such a transversal always exists under the additional hypothesis that rank(G) 3. To achieve this, the authors introduced a new technique called shifting boxes.…”

“…This description of Γ G H,H gave rise to a combinatorial model of coset intersections known as 'chessboards'. Chessboards provide an approach to answering versions of Question 1.3, as indeed was done in [3], and which we do throughout this paper. In particular, we use these techniques, as done in [3,Theorem 3.11], to relax the hypothesis of rank(G) 3, up to rank(G) 4.…”

“…Chessboards provide an approach to answering versions of Question 1.3, as indeed was done in [3], and which we do throughout this paper. In particular, we use these techniques, as done in [3,Theorem 3.11], to relax the hypothesis of rank(G) 3, up to rank(G) 4. That is, we show as the first main result of this paper that: Theorem 1.4.…”