We investigate the extent to which the exchange relation holds in finite groups G. We define a new equivalence relation ≡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 ≡(r) m of equivalence relations by saying that x ≡(r) m y if each can be substituted for the other in any r-element generating set. The relations ≡(r) m become finer as r increases, and we define a new group invariant ψ(G) to be the value of r at which they stabilise to ≡m. Remarkably, we are able to prove that if G is soluble, then ψ(G) ∈ {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 ψ(G) = d(G). For insoluble G, we show that d(G) ≤ ψ(G) ≤ d(G) + 5. However, we know of no examples of groups G for which ψ(G) > d(G) +1. As an application, we look at the generating graph Γ(G) of G, whose vertices are the elements of G, the edges being the 2-element generating sets. Our relation ≡(2) m enables us to calculate Aut(Γ(G)) for all soluble groups G of nonzero spread and to give detailed structural information about Aut(Γ(G)) in the insoluble case.
Generating sets of finite groups
Lucchini, Andrea
;RONEY-DOUGAL, COLVA MARY
2018
Abstract
We investigate the extent to which the exchange relation holds in finite groups G. We define a new equivalence relation ≡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 ≡(r) m of equivalence relations by saying that x ≡(r) m y if each can be substituted for the other in any r-element generating set. The relations ≡(r) m become finer as r increases, and we define a new group invariant ψ(G) to be the value of r at which they stabilise to ≡m. Remarkably, we are able to prove that if G is soluble, then ψ(G) ∈ {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 ψ(G) = d(G). For insoluble G, we show that d(G) ≤ ψ(G) ≤ d(G) + 5. However, we know of no examples of groups G for which ψ(G) > d(G) +1. As an application, we look at the generating graph Γ(G) of G, whose vertices are the elements of G, the edges being the 2-element generating sets. Our relation ≡(2) m enables us to calculate Aut(Γ(G)) for all soluble groups G of nonzero spread and to give detailed structural information about Aut(Γ(G)) in the insoluble case.File | Dimensione | Formato | |
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