Wydawnictwa / Czasopisma IMPAN / Bulletin of the Polish Academy of Sciences. Mathematics / Wszystkie zeszyty

Streszczenie

Let $G$ be an infinite abelian $p$-group and let $K$ be a field of the first kind with respect to $p$ of characteristic different from $p$ such that $s_p(K) = {\mathbb N}$ or $s_p(K)={\mathbb N}\cup\{0\}$. The main result of the paper is the computation of the Ulm–Kaplansky functions of the factor group $S(KG)/G$ of the normalized Sylow $p$-subgroup $S(KG)$ in the group ring $KG$ modulo $G$. We also characterize the basic subgroups of $S(KG)/G$ by proving that they are isomorphic to $S(KB)/B$, where $B$ is a basic subgroup of $G$.