부산대학교 도서관

If $X$ is a group, define $\scr R(X)$ to be the intersection of all normal subgroups of $X$ of finite index. Let $\scr R_\mu(X)$ be $\scr R(\scr R_{\mu-1}(X))$ if $\mu$ has a predecessor and $\scr R(X)=\bigcap_{\lambda<\mu}\scr R_\lambda(X)$ if $\mu$ is a limit ordinal. This gives a descending chain of subgroups, and the first ordinal $\lambda$ for which $\scr R_\lambda(X)=1$ is called by the author the ``length of the chain of higher residua of $X$''. We will refer to it as the ``$\scr R$-length of $X$'' for brevity. \par The main result concerns the relation between the $\scr R$-length of $G$, an abelian $p$-group, and the $\scr R$-length of $A(G)$, its group of automorphisms. For such groups $G$, $\scr R(G)=p^\omega G$. Therefore, let the Ulm type of $G$ be $\tau$, and express $\tau$ in normal expansion form: $\tau=\omega^\alpha\cdot n+\omega^{\alpha_1}\cdot n_1+\cdots+\omega^{\alpha_k}\cdot n_k$, where $\alpha>\alpha_1>\cdots>\alpha_k$ are ordinals and $n,n_1,\cdots,n_k$ are positive integers. Let $n<2^m$ for some integer $m$. Then the author gives the following theorem: If $\lambda$ is the least ordinal such that $\scr R_\lambda(A(G))=1$ then $\lambda\leq\omega\alpha+m+2$. \par \{The reader may find this paper easier to read if he observes that $p^\alpha$ and $\scr R$ are radicals [see R. J. Nunke, Math. Z. {\bf 101} (1967), 182--212; MR0218452 (36 \#1538)]. Then Lemmas 2.1 and 3.1 are obvious.\} \par The author also explains the connection between $\scr R$-length and the shortest length of chains of subgroups with residually finite factors.