@article{2d4aaa75bf1b4947b926df827035c4ce,
title = "Finite groups satisfying the independence property",
abstract = "We say that a finite group G satisfies the independence property if, for every pair of distinct elements x and y of G, either {x,y} is contained in a minimal generating set for G or one of x and y is a power of the other. We give a complete classification of the finite groups with this property, and in particular prove that every such group is supersoluble. A key ingredient of our proof is a theorem showing that all but three finite almost simple groups H contain an element s such that the maximal subgroups of H containing s, but not containing the socle of H, are pairwise non-conjugate.",
keywords = "Generating sets, simple groups, supersoluble groups",
author = "Freedman, {Saul D.} and Andrea Lucchini and Daniele Nemmi and Roney-Dougal, {Colva M.}",
note = "Funding Information: We are grateful to Peter Cameron for helpful discussions, and in particular for suggesting the problem of classifying the groups satisfying the rank-independence property; and to anonymous referees for helpful comments. The first author was supported by a St Leonard's International Doctoral Fees Scholarship and a School of Mathematics & Statistics PhD Funding Scholarship at the University of St Andrews. The fourth author would like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programme Groups, Representations and Applications: New perspectives, where work on this paper was undertaken. This work was supported by EPSRC Grant No. EP/R014604/1, and also partially supported by a grant from the Simons Foundation. Publisher Copyright: {\textcopyright} 2023 World Scientific Publishing Company.",
year = "2023",
month = may,
day = "1",
doi = "10.1142/S021819672350025X",
language = "English",
volume = "33",
pages = "509--545",
journal = "International Journal of Algebra and Computation",
issn = "0218-1967",
publisher = "World Scientific Publishing",
number = "3",
}