Abstract
Given a group G, we write xG for the conjugacy class of G containing the element x. A famous theorem of B. H. Neumann states that if G is a group in which all conjugacy classes are finite with bounded size, then the derived group G′ is finite. We establish the following result.
Let n be a positive integer and K a subgroup of a group G such that ∣xG∣ ≤ n for each x ∈ K. Let H = 〈KG〉 be the normal closure of K. Then the order of the derived group H′ is finite and n-bounded.
Some corollaries of this result are also discussed.
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Dedicated to Andrea Lucchini on the occasion of his 60th birthday
This research was supported by the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), and Fundação de Apoio à Pesquisa do Distrito Federal (FAPDF), Brazil.
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Acciarri, C., Shumyatsky, P. A stronger form of Neumann’s BFC-theorem. Isr. J. Math. 242, 269–278 (2021). https://doi.org/10.1007/s11856-021-2133-1
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DOI: https://doi.org/10.1007/s11856-021-2133-1