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Keywords:
automorphism; Boolean ring; divisible Abelian group; Galois ring; homococyclic group; injective hull; $p$-component; quasi-cyclic group; uniserial ring
automorphism; Boolean ring; divisible Abelian group; Galois ring; homococyclic group; injective hull; $p$-component; quasi-cyclic group; uniserial ring
Summary:
Let $G$ be an Abelian group. Then $G$ is called a $T_1'$ group provided that for every pair of isomorphic subgroups $H$ and $K$ of $G$ and every isomorphism $\varphi \colon H\to K$, $\varphi $ can be extended to an automorphism of $G$. Let us call $G$ a weakly $T_1'$ group if for any elements $a$ and $b$ of $G$ of the same order, there is an automorphism $\varphi \colon G\to G$ such that $\varphi (a)=b$. We determine both classes of groups, correcting a result in the literature. We then study several analogues for rings.
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