×

zbMATH — the first resource for mathematics

3-rewritable nilpotent 2-groups of class 2. (English) Zbl 1088.20007
A group is 3-rewritable if for any collection of three elements at least two of the six possible products are equal. Groups of odd order with this property have a commutator subgroup of order at most 5. – The authors show in general that for these groups \(G\) the subgroup \([G',G^2]\) is of exponent dividing 480. For 2-groups the authors consider especially extensions of Abelian groups by cyclic groups. They show: If \(G_3=1\), then \(G\) is 3-rewritable iff \(|\langle x,y,z\rangle'|\leq 4\) for every triple \(x,y,z\) (Theorem B); if there is \(A\) with \(G'\subseteq A=C(A)\subset G\) such that \(G/A\) is not elementary Abelian, then \(G_5=1\) (Theorem A).

MSC:
20D15 Finite nilpotent groups, \(p\)-groups
20F14 Derived series, central series, and generalizations for groups
20F12 Commutator calculus
20F05 Generators, relations, and presentations of groups
20D60 Arithmetic and combinatorial problems involving abstract finite groups
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Blyth R. D., J. Algebra 116 pp 506– (1988) · Zbl 0647.20033
[2] Blyth R. D., J. Algebra 119 pp 246– (1988) · Zbl 0663.20036
[3] Blyth R. D., Q3 Arch. Math. Basel 78 (5) pp 337– (2002) · Zbl 1011.20024
[4] Blyth R. D., Q5 Comm. Algebra 23 (6) pp 2171– (1995) · Zbl 0831.20027
[5] Curzio M., Atti Acc. Lincei Rend. Sem. Mat. Fis. Nat. 74 pp 136– (1983)
[6] Curzio M., Arch. Math. 44 pp 385– (1985) · Zbl 0544.20036
[7] Longobardi P., Illinois Journal of Mathematics 35 (2) pp 198– (1991)
[8] Longobardi P., Rend. Semin. Mat. Univ. Padova 93 pp 7– (1995)
[9] Maj M., J. Algebra 136 (1) pp 86– (1991) · Zbl 0721.20022
[10] Maj M., Can. J. Math. 42 (6) pp 1053– (1990) · Zbl 0727.20027
[11] The GAP GroupGAP – Groups Algorithms and ProgrammingVersion 4.3; 2002 (http://www.gap-system.org).
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.