CONGRUENCES ON \((n,m)\)-GROUPS - Janeva, Biljana (Matematichki Bilten, 1999)
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 \[ x_jy_j^-1K,j=1,n &x_1… x_n=a_1… a_m,y_1… y_n=b_1… b_m a_ib_i^-1K, \] for all \(i=1,m\), there exists a congruence \(α\) on the \((n,m)\)-group \(Q\), such that the corresponding invariant subgroup of \(Q^V\) is exactly \(K\).