Hassanabadi, A. Mohammadi A property equivalent to permutability for groups. (English) Zbl 0926.20021 Rend. Semin. Mat. Univ. Padova 100, 137-142 (1998). Let \(m\) and \(n\) be positive integers. The author calls a group \(G\) restricted \((m,n)\)-permutable if \[ X_1X_2\cdots X_n\cap\bigcup_{\sigma\in S_m\setminus 1} X_{\sigma(1)}X_{\sigma(2)}\cdots X_{\sigma(n)} \] is non-empty for all \(m\)-element subsets \(X_i\) of \(G\). When \(m=1\), this is the same as \(n\)-permutable, i.e., given \(x_1,x_2,\dots,x_n\in G\), there is a permutation \(\sigma\neq 1\) such that \(x_1x_2\cdots x_n=x_{\sigma(1)}x_{\sigma(2)}\cdots x_{\sigma(n)}\).The following main result is proved. Theorem: If a group \(G\) is restricted \((m,n)\)-permutable, then \(G\) is finite-by-abelian-by-finite. When \(m=1\), this is a theorem of Curzio, Longobardi, Maj and the reviewer (which is used in the proof). Reviewer: D.J.S.Robinson (Urbana) Cited in 2 Documents MSC: 20F19 Generalizations of solvable and nilpotent groups 20F05 Generators, relations, and presentations of groups Keywords:restricted permutable groups; permutation properties; finite-by-Abelian-by-finite groups PDFBibTeX XMLCite \textit{A. M. Hassanabadi}, Rend. Semin. Mat. Univ. Padova 100, 137--142 (1998; Zbl 0926.20021) Full Text: Numdam EuDML References: [1] R.D. Blyth , Rewriting products of group elements, I , J. Algebra , 116 ( 1988 ), pp. 506 - 521 . MR 953167 | Zbl 0647.20033 · Zbl 0647.20033 [2] R.D. Blyth , Rewriting products of group elements, II , J. Algebra , 119 ( 1988 ), pp. 246 - 259 . MR 971358 | Zbl 0663.20036 · Zbl 0663.20036 [3] M. Curzio - P. LONGOBARDI - M. MAJ - D. J. S. ROBINSON, A permutational property of groups , Arch. Math. ( Basel ), 44 ( 1985 ) pp. 385 - 389 . MR 792360 | Zbl 0544.20036 · Zbl 0544.20036 [4] P. Longobardi - M. MAJ - S. E. STONEHEWER, The classification of groups in which every product of four elements can be reordered , Rend. Sem. Mat. Univ. Padova , 93 ( 1995 ), pp. 7 - 26 . Numdam | MR 1354348 | Zbl 0838.20038 · Zbl 0838.20038 [5] M. Maj - S.E. Stonehewer , Non-nilpotent groups in which every product of four elements can be reordered , Canadian J. , 42 ( 1990 ), pp. 1053 - 1066 . MR 1099457 | Zbl 0727.20027 · Zbl 0727.20027 [6] A. Mohammadi Hassanabadi - Akbar Rhemtulla , Criteria for commutativity in large groups , Canda. Math. Bull. , Vol. XX ( Y ) ( 1998 ), pp. 1 - 6 . MR 1618947 | Zbl 0908.20022 · Zbl 0908.20022 [7] B.H. Neumann , Groups covered by permutable subsets , J. London Math. Soc. , 29 ( 1954 ), pp. 236 - 248 . MR 62122 | Zbl 0055.01604 · Zbl 0055.01604 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.