About group topologies of the primary Abelian group of finite period which coincide on a~subgroup and on the factor group

V. I. Arnautov

Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 2 (2009), pp. 19-28

Résumé

Let $G$ be any Abelian group of the period $p^n$ and $G_1=\{g\in G\mid pg=0\}$, $G_2=\{g\in G\mid p^{n-1}g=0\}$. If $\tau$ and $\tau'$ are a metrizable, linear group topologies such that $G_2$ is a closed subgroup in each of topological groups $(G,\tau)$ and $(G,\tau')$, then $\tau|_{G_2}=\tau'|_{G_2}$ and $(G,\tau)/G_1=(G,\tau')/G_1$ if and only if there exists a group isomorphism $\varphi\colon G\to G$ such that the following conditions are true: 1. $\varphi(G_2)=G_2$; 2. $g-\varphi(g)\in G_1$ for any $g\in G$; 3. $\varphi\colon (G,\tau)\to(G,\tau')$ is a topological isomorphism.