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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).

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
Full Text: DOI
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