CONGRUENCES ON \(n\)-GROUPS

Abstract

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
\[
H_\alpha\subseteq\{a_1\cdots a_{n-1}\mid a_\nu\in Q\}.
\]
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.