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TRANSVERSALS AS GENERATING SETS IN FINITELY GENERATED GROUPS

Part of: Special aspects of infinite or finite groups Structure and classification of infinite or finite groups

Published online by Cambridge University Press:  19 August 2015

JACK BUTTON ,
MAURICE CHIODO  and
MARIANO ZERON-MEDINA LARIS
JACK BUTTON
Affiliation:
Selwyn College, Cambridge, Grange Road, CambridgeCB3 9DQ, UK email [email protected]
MAURICE CHIODO*
Affiliation:
Mathematics Department, University of Neuchâtel, Rue Emile-Argand 11, CH-2000 Neuchâtel, Switzerland email [email protected]
MARIANO ZERON-MEDINA LARIS
Affiliation:
31 Mariner’s Way, CambridgeCB4 1BN, UK email [email protected]
*
✉[email protected]
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Abstract

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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 $n\leq 3$, 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 $3\cdot 2^{n-1}$, and $H$ contains no primitive elements of $G$, then $H$ is normal in $G$ and $G/H\cong C_{2}^{n}$.

Keywords

transversalsgenerating setsfinite index subgroupsprimitive elements

MSC classification

Primary: 20F05: Generators, relations, and presentations
Secondary: 20E99: None of the above, but in this section
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 93 , Issue 1 , February 2016 , pp. 47 - 60
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

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