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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Generating sets of finite groups


Authors: Peter J. Cameron, Andrea Lucchini and Colva M. Roney-Dougal
Journal: Trans. Amer. Math. Soc. 370 (2018), 6751-6770
MSC (2010): Primary 20D60; Secondary 20D10, 20D05
DOI: https://doi.org/10.1090/tran/7248
Published electronically: April 4, 2018
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Abstract: We investigate the extent to which the exchange relation holds in finite groups $ G$. We define a new equivalence relation $ \equiv _{\mathrm {m}}$, where two elements are equivalent if each can be substituted for the other in any generating set for $ G$. We then refine this to a new sequence $ \equiv _{\mathrm {m}}^{(r)}$ of equivalence relations by saying that $ x \equiv _{\mathrm {m}}^{(r)}y$ if each can be substituted for the other in any $ r$-element generating set. The relations $ \equiv _{\mathrm {m}}^{(r)}$ become finer as $ r$ increases, and we define a new group invariant $ \psi (G)$ to be the value of $ r$ at which they stabilise to $ \equiv _{\mathrm {m}}$.

Remarkably, we are able to prove that if $ G$ is soluble, then $ \psi (G) \in \{d(G),$
$ d(G) +1\}$, where $ d(G)$ is the minimum number of generators of $ G$, and to classify the finite soluble groups $ G$ for which $ \psi (G) = d(G)$. For insoluble $ G$, we show that $ d(G) \leq \psi (G) \leq d(G) + 5$. However, we know of no examples of groups $ G$ for which $ \psi (G) > d(G) + 1$.

As an application, we look at the generating graph $ \Gamma (G)$ of $ G$, whose vertices are the elements of $ G$, the edges being the $ 2$-element generating sets. Our relation $ \equiv _{\mathrm {m}}^{(2)}$ enables us to calculate $ \textup {Aut}(\Gamma (G))$ for all soluble groups $ G$ of nonzero spread and to give detailed structural information about $ \textup {Aut}(\Gamma (G))$ in the insoluble case.


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Additional Information

Peter J. Cameron
Affiliation: Mathematical Institute, University of St Andrews, St Andrews, Fife KY16 9SS, Scotland
Email: pjc20@st-andrews.ac.uk

Andrea Lucchini
Affiliation: Dipartimento di Matematica, Università degli studi di Padova, Via Trieste 63, 35121 Padova, Italy
Email: lucchini@math.unipd.it

Colva M. Roney-Dougal
Affiliation: Mathematical Institute, University of St Andrews, St Andrews, Fife KY16 9SS, Scotland
Email: colva.roney-dougal@st-andrews.ac.uk

DOI: https://doi.org/10.1090/tran/7248
Keywords: Finite group, generation, generating graph
Received by editor(s): September 19, 2016
Received by editor(s) in revised form: January 9, 2017
Published electronically: April 4, 2018
Additional Notes: The second and third authors were supported by Università di Padova (Progetto di Ricerca di Ateneo: Invariable generation of groups).
Article copyright: © Copyright 2018 American Mathematical Society

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