Sylow structure of an idempotent \(n\)-ary group. (Q2784890)

An element of an \(n\)-ary group \(\langle A,[\;]\rangle\) is called idempotent if \([\underbrace{a,\dots,a}_n]=a\). It is shown that any finite idempotent \(n\)-ary group \(\langle A,[\;]\rangle\) of order \(p^km\) where \((p,m)=1\), \(p\) and \(n-1\) are prime, can be written as a union \(A=\bigcup_{i=1}^mP_i\), \(P_i\cap P_j=\emptyset\) (\(i\not=j\)) and \(\langle P_1,[\;]\rangle ,\dots,\langle P_m,[\;]\rangle\) are the only semiinvariant \(p\)-Sylow \(n\)-ary subgroups. It is also proved that any \(n\)-ary group \(\langle A,[\;]\rangle\) of order \(p_1^{\alpha_1}\cdots p_k^{\alpha_k}\) can be decomposed into the \(a\)-direct product of its \(p_i\)-Sylow \(n\)-ary subgroups \(\langle A(p_i),[\;]\rangle\) (the notion of an \(a\)-direct product was earlier introduced by S.~A.~Rusakov).