CONGRUENCES ON \((n,m)\)-GROUPS

Abstract

In this paper we give a generalization of some definitions and properties of congruences on \(n\)-groups given in [2]. Congruences on \((n,m)\)-groups are defined as congruences of the corresponding component algebra, and as kernel of a homomorphism, and a connection between these two definitions is given. Also, it is shown that for each congruence of an \((n,m)\)-group \(Q\) there exists an invariant subgroup of its universal covering group \(Q^V\) of \(Q\), that is a subset of \(Q_{m+p}\), where \(m+p=sk\), \(k=n-m>0\). Conversely, for each invariant subgroup \(K\) of \(Q^V\), which is a subset of \(Q_{m+p}\) and satisfies the condition
\[
x_jy_j^{-1}\in K,\quad j=\overline{1,n}
\quad\&\quad
x_1\cdots x_n=a_1\cdots a_m,\quad
y_1\cdots y_n=b_1\cdots b_m
\Rightarrow a_ib_i^{-1}\in K,
\]
for all \(i=\overline{1,m}\), there exists a congruence \(\alpha\) on the \((n,m)\)-group \(Q\), such that the corresponding invariant subgroup of \(Q^V\) is exactly \(K\).