Generating sets of finite groups

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 examples 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.