부산대학교 도서관

This work answers a question of R. Baer, namely, for an Abelian torsion group $G$, what group property imposed on $A(G)$, the automorphism group of $G$, is necessary and sufficient for every primary component of $G$ to be Artinian? (Baer showed [Trans. Amer. Math. Soc. {\bf 79} (1955), 521--540; MR0071425 (17,125a)] that $G$ is Artinian if and only if every torsion subgroup of $A(G)$ is finite.) \par First the following theorem is proved, in which ``$p'$-subgroup'' means a subgroup with no elements of order $p$. Theorem: Let $G$ be an Abelian $p$-group for $p>3$ and $\Gamma$ a normal $p'$-subgroup of $A(G)$; then $\Gamma$ is contained in the center of $A(G)$. \par A group is said to be residually finite if the intersection of all subgroups of finite index is just the identity element. For a subgroup $\Gamma$ of $A(G)$, $c\Gamma$ denotes its centralizer in $A(G)$. The main theorem of the paper is that every primary component of the Abelian torsion group $G$ is Artinian if and only if $A(G)$ is residually finite and $A(G)/c\Gamma$ is finite for every primary normal subgroup $\Gamma$ of $A(G)$. \par References to results from other works are carefully footnoted. A serious reader would need particularly to refer to Baer's paper mentioned above and to another paper of the author's [Pacific J. Math. {\bf 35} (1970), 127--139; MR0271208 (42 \#6091)].