Clones of 2-step nilpotent groups
The clone of an algebra \(A\) in a language \(L\) is the collection Clo\((A)\) of operations on \(A\) that are the interpretations of the \(L\)-terms. The members of Clo\((A)\) are called term operations of \(A\). A \(k\)-ary algebraic relation on \(A\) is a subalgebra of \(A^{k}\). In \textit{K. A. Kearnes} and \textit{Á. Szendrei}'s paper [Algebra Univers. 54, No.~1, 23--52 (2005; Zbl 1114.20014)] it is proved that if \(G\) is a finite group with abelian Sylow subgroups then Clo\((G)\) is determined by the \(3\)-ary algebraic relations of \(G\), and if \(G\) is a finite nilpotent group then Clo\((G)\) is determined by the \(d\)-ary algebraic relations of \(G\) for \(d=\left| G\right| ^{[G:Z(G)]-1}\). In this paper the authors refine a special case of the last result to show that if \(G\) is a finite \(2\)-step nilpotent group, then Clo\((G)\) is determined by the \(4\)-ary algebraic relations of \(G\). (It is shown in K. A. Kearnes and Á. Szendrei's paper [loc. cit.] that if \(Q\) is the quaternion group, then Clo\((Q)\) is not determined by the 3-ary algebraic relations of \(Q\)).
scientific article; zbMATH DE number 5508761