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Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, 2009, Number 2, Pages 19–28
(Mi basm224)
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Research articles
About group topologies of the primary Abelian group of finite period which coincide on a subgroup and on the factor group
V. I. Arnautov Institute of Mathematics and Computer Science, Academy of Sciences of Moldova, Chicşinau, Moldova
Abstract:
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.
Keywords and phrases:
topological group, natural homomorphism, topological isomorphism, subgroup of topological group, factor group of topological group, basis of neighborhoods of zero.
Received: 12.05.2009
Citation:
V. I. Arnautov, “About group topologies of the primary Abelian group of finite period which coincide on a subgroup and on the factor group”, Bul. Acad. Ştiinţe Repub. Mold. Mat., 2009, no. 2, 19–28
Linking options:
https://www.mathnet.ru/eng/basm224 https://www.mathnet.ru/eng/basm/y2009/i2/p19
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