zbMATH — the first resource for mathematics

Characterization of Abelian-by-cyclic \(3\)-rewritable groups. (English) Zbl 1138.20035
Summary: Let \(n\) be an integer greater than 1. A group \(G\) is said to be \(n\)-rewritable (or a \(Q_n\)-group) if for every \(n\) elements \(x_1,\dots,x_n\) in \(G\) there exist distinct permutations \(\sigma\) and \(\tau\) in \(S_n\) such that \(x_{\sigma(1)}x_{\sigma(2)}\cdots x_{\sigma(n)}=x_{\tau(1)}x_{\tau(2)}\cdots x_{\tau(n)}\). In this paper, we completely characterize Abelian-by-cyclic 3-rewritable groups: they turn out to have an Abelian subgroup of index 2 or the size of derived subgroups is less than 6. In this paper, we also prove that \(G/F(G)\) is an Abelian group of finite exponent dividing 12, where \(F(G)\) is the Fitting subgroup of \(G\).

20F05 Generators, relations, and presentations of groups
20E07 Subgroup theorems; subgroup growth
20F16 Solvable groups, supersolvable groups
Full Text: EuDML
[1] A. ABDOLLAHI - A. MOHAMMADI HASSANABADI, 3-rewritable nilpotent 2-groups of class 2, to appear in Comm. Algebra, 32 (2004). Zbl1088.20007 MR2149067 · Zbl 1088.20007
[2] M. BIANCHI - R. BRANDL - A. GILLIO BERTA MAURI, On the 4-permutational property, Arch. Math. (Basel), 48, No. 4, (1987), pp. 281-285. Zbl0623.20022 MR884558 · Zbl 0623.20022
[3] R. D. BLYTH, Odd order groups with the rewriting property Q3 , Arch. Math. (Basel), 78, No. 5 (2002), pp. 337-344. Zbl1011.20024 MR1903666 · Zbl 1011.20024
[4] R. D. BLYTH, Rewriting products of group elements-II, J. Algebra, 119 (1988), pp. 246-259. Zbl0663.20036 MR971358 · Zbl 0663.20036
[5] R. D. BLYTH, Rewriting products of group elements-I, J. Algebra, 116 (1988), pp. 506-521. Zbl0647.20033 MR953167 · Zbl 0647.20033
[6] R. D. BLYTH - D. J. S. ROBINSON, Semisimple groups with the rewriting property Q5, Comm. Algebra, 23, No. 6 (1995), pp. 2171-2180. Zbl0831.20027 MR1327132 · Zbl 0831.20027
[7] R. BRANDL, Zur theorie der untergruppenabgeshlossenen formationen: Endlich varietäten, J. Algebra, 73 (1981), pp. 1-22. Zbl0484.20012 MR641629 · Zbl 0484.20012
[8] M. CURZIO - P. LONGOBARDI - M. MAJ, Su di un problema combinatorio in teoria dei gruppi, Atti Acc. Lincei Rend. Sem. Mat. Fis. Nat., 74 (1983), pp. 136-142. Zbl0528.20031 MR739397 · Zbl 0528.20031
[9] M. CURZIO - P. LONGOBARDI - M. MAJ - D. J. S. ROBINSON, On a permutational properties of groups, Arch. Math. (basel), 44 (1985), pp. 385-389. Zbl0544.20036 MR792360 · Zbl 0544.20036
[10] P. LONGOBARDI - M. MAJ - S. STONEHEWER, The classification of groups in which every product of four elements can be reordered, Rend. Semin. Mat. Univ. Padova, 93 (1995), pp. 7-26. Zbl0838.20038 MR1354348 · Zbl 0838.20038
[11] P. LONGOBARDI - S. E. STONEHEWER, Finite 2-groups of class 2 in which every product of four elements can be reordered, Illinois Journal of Mathematics, 35, No. 2 (1991), pp. 198-219. Zbl0698.20013 MR1091438 · Zbl 0698.20013
[12] M. MAJ, On the derived length of groups with some permutational property, J. Algebra 136, No. 1 (1991), pp. 86-91. Zbl0721.20022 MR1085122 · Zbl 0721.20022
[13] M. MAJ - S. E. STONEHEWER, Non-nilpotent groups in which every product of four elements can be reordered, Can. J. Math., 42, No. 6 (1990), pp. 1053-1066. Zbl0727.20027 MR1099457 · Zbl 0727.20027
[14] D. J. S. ROBINSON, A course in the theory of groups, 2nd ed., Berlin-New York, 1995. Zbl0836.20001 MR1357169 · Zbl 0836.20001
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.