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Permutability of centre-by-finite groups

Brunetto Piochi (1989)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Let G be a group and m be an integer greater than or equal to 2 . G is said to be m -permutable if every product of m elements can be reordered at least in one way. We prove that, if G has a centre of finite index z , then G is ( 1 + [ z / 2 ] ) -permutable. More bounds are given on the least m such that G is m -permutable.

Su di un problema combinatorio in teoria dei gruppi

Mario Curzio, Patrizia Longobardi, Mercede Maj (1983)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Let G be a group and n an integer ≥ 2 . We say that G has the n -permutation property ( G ∈ P n ) if, for any elements x 1 , x 2 , ⋯ , x n in G , there exists some permutation σ of { 1 , 2 , ⋯ , n } , σ ≠ i ⁢ d . such that x 1 , x 2 , ⋯ , x n = x σ ⁢ ( 1 ) , x σ ⁢ ( 2 ) , ⋯ , x σ ⁢ ( n ) . We prouve that every group G ∈ P n is an FC-nilpotent group of class ≤ n - 1 , and that a finitely generated group has the n -permutation property (for some n ) if, and only if, it is abelian by finite. We prouve also that a group G ∈ P 3 if, and only if, its derived subgroup has order at most 2.

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