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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).

##### 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
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