CONGRUENCES ON \(n\)-GROUPS - Janeva, Biljana (Matematichki Bilten, 1995)
A description of congruences on polyadic groups is given in this paper. Namely, if \(Q=(Q,[\,])\) is an \(n+1\)-group, and \(Q^\wedge\) is its universal covering group, then each congruence \(\alpha\) of \(Q\) can be characterized by an invariant subgroup \(H_\alpha\) of \(Q^\wedge\) such that\[H_\alpha\subseteq\{a_1\cdots a_{n-1}\mid a_ u\in Q\}.\]Also, if \(a\in Q\) and \(*\) is an operation on \(Q\) defined by\[x*y=[xa^{n-2}y],\]then \((Q,*)\) is a group, and each congruence \(\alpha\) of \(Q\) is characterized by an invariant subgroup \(K\) of \((Q,*)\), such that for each \(x\in Q\)\[[xa^{n-2}K]=[a^{n-2}Kx].\]It is also shown that it may happen to have an \(n\)-group \(Q\) and a congruence \(\alpha\) on \(Q\) such that neither of the \(\alpha\)-equivalence classes is an \(n\)-subgroup of \(Q\), and necessary and sufficient conditions are given under which such classes do exist.In the last section of this paper new and shorter proofs of some propositions of [5] are given using the universal covering group.