On the properties of Cartesian powers of coset groups and polyadic groups of matrices. (Q2901303)

Let \(A\) be a groupoid (a set with a single binary operation). Let \(k\geq 2\) and \(\sigma\) be a permutation of the set \(\{1,\dots,k\}\). The following binary operation is considered on the set \(A^K\): NEWLINE\[NEWLINE\mathbf x\overset\sigma\circ\mathbf y=(x_1,\dots,x_k)\overset\sigma\circ(y_1,\dots,y_k)=(x_1\sigma(y_1),\dots,x_k\sigma(y_k))NEWLINE\]NEWLINE so as an \(l\)-ary operation for \(l\geq 2\): NEWLINE\[NEWLINE[\mathbf x_1,\mathbf x_2,\dots,\mathbf x_l]_{l,\sigma,k}=\mathbf x_1\overset\sigma\circ(\mathbf x_2\overset\sigma\circ(\cdots(\mathbf x_{l-2}\overset\sigma\circ(\mathbf x_{l-1}\overset\sigma\circ\mathbf x_l))\cdots)).NEWLINE\]NEWLINE Let \(A\) be a group and let \(\sigma\) satisfy the condition \(\sigma^l=\sigma\). Then \(\langle A^k,[\;]_{l,\sigma,k}\rangle\) is an \(l\)-ary group. Let in addition \(B\) be a normal subgroup of \(A\) such that the factor group \(A/B\) is cyclic and has order a divisor of \(l-1\). In this case for any element \(H\) of the factor group \(A/B\), the Cartesian power \(H^k\) is closed relatively to the \(l\)-ary operation \([\;]_{l,\sigma,k}\). The authors study the properties of this \(l\)-ary operation on Cartesian powers of conjugate group classes of the group \(A\) associated to \(B\).