1,721,248 research outputs found
Finite soluble groups satisfying the swap conjecture
No full textFor a d-generated finite group G, we consider the graph Δd(G) (swap graph) in which the vertices are the ordered generating d-tuples and in which two vertices (x1,...,xd) and (y1,...,yd) are adjacent if and only if they differ only by one entry. It was conjectured by Tennant and Turner that Δd(G) is a connected graph. We prove that this conjecture is true if G is a soluble group satisfying some extra conditions, for example if the derived subgroup of G has odd order or is nilpotent
Invariable generation of iterated wreath products of cyclic groups
Full text linkGiven a sequence Ci of cyclic groups of prime orders, let Γ be the inverse limit of the iterated wreath products Cm≀ ⋯ ≀ C2≀ C1. We prove that the profinite group Γ is not topologically finitely invariably generated
The X-Dirichlet polynomial of a finite group
Full text linkFor a finite group G and a subset X of G let P(G,X,t) denote the probability that t random elements of G together with X generate G. Then, as discovered by P. Hall, the function P(G,X,t) can be written as a Dirichlet polynomial and in this way can be defined for all complex values of t. As in the case of X=∅, the main result is that the factors of P(G,X,t) obtained in this way are independent of the choice of the chief series. We shall describe the Dirichlet polynomials that arise in this kind of factorization
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