On \(t\)-uniform groups
A group \(G\) is said to be hyperrational if it has an ascending series \(1 = G_ 0 \leq G_ 1 \leq \cdots \leq G_ \alpha\) such that, for every \(\beta < \alpha\), \(G_ \beta\) is an isolated normal subgroup of \(G_{\beta + 1}\) and the factor group \(G_{\beta + 1}/G_ \beta\) is isomorphic to a subgroup of the additive group of rational numbers. If the above series can be chosen to be finite, the group \(G\) is called polyrational. Moreover, the hyperrational group \(G\) is called \(t\)-uniform if all the factor groups \(G_{\beta + 1}/G_ \beta\) have the same type \(t\). Among other results, it is proved that if \(G\) is a non-abelian \(t\)- uniform Engel group, then the type \(t\) is idempotent. It follows that if \(G\) is a \(t\)-uniform polyrational non-abelian \(R\)-group, then \(t\) is idempotent (here \(G\) is said to be an \(R\)-group if for every positive integer \(n\) and for all elements \(x, y \in G\) such that \(x^ n = y^ n\) we have \(x = y\)). Moreover, some results on homogeneous \(t\)-uniform groups and on normal homogeneous subgroups are obtained.
scientific article; zbMATH DE number 493391
scientific article