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Nilpotent groups of class three and braces. (English) Zbl 1345.16039
J. C. Ault and J. F. Watters [Am. Math. Mon. 80, 48-52 (1973; Zbl 0251.16009)] and the authors in a previous article [Commun. Math. Phys. 327, No. 1, 101-116 (2014; Zbl 1287.81062)] have shown that every finitely generated nilpotent group of class 2 is the adjoint group of a radical ring. For any radical ring, the identity map is a 1-cocycle, connecting the adjoint group with the additive group. More generally, a bijective 1-cocycle is equivalent to a structure which is now called a brace. In the paper under review, braces with nilpotent adjoint group of class 3 and odd order are constructed. It is shown, for example, that every nilpotent group of class 3 and odd order is a homomorphic image of the adjoint group of such a brace.

##### MSC:
 16T25 Yang-Baxter equations 81R50 Quantum groups and related algebraic methods applied to problems in quantum theory 16N40 Nil and nilpotent radicals, sets, ideals, associative rings 20D15 Finite nilpotent groups, $$p$$-groups
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