ORCID Profile
0000-0002-0532-3349
Current Organisation
University of St Andrews
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Publisher: Elsevier BV
Date: 03-2020
Publisher: American Mathematical Society (AMS)
Date: 04-04-2018
DOI: 10.1090/TRAN/7248
Abstract: We investigate the extent to which the exchange relation holds in finite groups G G . We define a new equivalence relation ≡ m \\equiv _{\\mathrm {m}} , where two elements are equivalent if each can be substituted for the other in any generating set for G G . We then refine this to a new sequence ≡ m ( r ) \\equiv _{\\mathrm {m}}^{(r)} of equivalence relations by saying that x ≡ m ( r ) y x \\equiv _{\\mathrm {m}}^{(r)}y if each can be substituted for the other in any r r -element generating set. The relations ≡ m ( r ) \\equiv _{\\mathrm {m}}^{(r)} become finer as r r increases, and we define a new group invariant ψ ( G ) \\psi (G) to be the value of r r at which they stabilise to ≡ m \\equiv _{\\mathrm {m}} . Remarkably, we are able to prove that if G G is soluble, then ψ ( G ) ∈ { d ( G ) , \\psi (G) \\in \\{d(G), d ( G ) + 1 } d(G) +1\\} , where d ( G ) d(G) is the minimum number of generators of G G , and to classify the finite soluble groups G G for which ψ ( G ) = d ( G ) \\psi (G) = d(G) . For insoluble G G , we show that d ( G ) ≤ ψ ( G ) ≤ d ( G ) + 5 d(G) \\leq \\psi (G) \\leq d(G) + 5 . However, we know of no ex les of groups G G for which ψ ( G ) d ( G ) + 1 \\psi (G) d(G) + 1 . As an application, we look at the generating graph Γ ( G ) \\Gamma (G) of G G , whose vertices are the elements of G G , the edges being the 2 2 -element generating sets. Our relation ≡ m ( 2 ) \\equiv _{\\mathrm {m}}^{(2)} enables us to calculate A u t ( Γ ( G ) ) \\mathrm {Aut}(\\Gamma (G)) for all soluble groups G G of nonzero spread and to give detailed structural information about A u t ( Γ ( G ) ) \\mathrm {Aut}(\\Gamma (G)) in the insoluble case.
Publisher: Springer Science and Business Media LLC
Date: 21-08-2015
Publisher: World Scientific Pub Co Pte Ltd
Date: 29-04-2023
DOI: 10.1142/S021819672350025X
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 [Formula: see text] 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.
Publisher: The Electronic Journal of Combinatorics
Date: 29-01-2021
DOI: 10.37236/9802
Abstract: For a nilpotent group $G$, let $\\Xi(G)$ be the difference between the complement of the generating graph of $G$ and the commuting graph of $G$, with vertices corresponding to central elements of $G$ removed. That is, $\\Xi(G)$ has vertex set $G \\setminus Z(G)$, with two vertices adjacent if and only if they do not commute and do not generate $G$. Additionally, let $\\Xi^+(G)$ be the subgraph of $\\Xi(G)$ induced by its non-isolated vertices. We show that if $\\Xi(G)$ has an edge, then $\\Xi^+(G)$ is connected with diameter $2$ or $3$, with $\\Xi(G) = \\Xi^+(G)$ in the diameter $3$ case. In the infinite case, our results apply more generally, to any group with every maximal subgroup normal. When $G$ is finite, we explore the relationship between the structures of $G$ and $\\Xi(G)$ in more detail.
Location: United Kingdom of Great Britain and Northern Ireland
No related grants have been discovered for Colva Roney-Dougal.