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On diagonal locally SL-groups. (English) Zbl 1474.20087

Summary: Let \(\mathbb{N}\) be the set of natural numbers. Let \(\mathbb{F}\) be a field. In O. O. Bezushchak and V. I. Sushchans’kyi [Ukr. Math. J. 67, No. 10, 1457–1468 (2016; Zbl 1376.20053); translation from Ukr. Mat. Zh. 67, No. 10, 1299–1308 (2015)] we introduced a class of groups \(\mathrm{SL}^p_s(\mathbb{F})\) and \(\mathrm{GL}^p_s(\mathbb{F})\) of periodic infinite \(\mathbb{N}\times\mathbb{N}\)-matrices that correspond to a Steinitz number \(s\). In this paper we introduce a wider class of diagonal locally \(\mathrm{SL}\)-groups and \(\mathrm{GL}\)-groups and study their relations with locally matrix algebras. In particular, we show that every separable-diagonal locally \(\mathrm{SL}\)-group (respectively \(\mathrm{GL}\)-group) is isomorphic to a group \(\mathrm{SL}^p_s(\mathbb{F})\) (respectively \(\mathrm{GL}^p_s(\mathbb{F})\)).

MSC:

20G15 Linear algebraic groups over arbitrary fields
20F50 Periodic groups; locally finite groups

Citations:

Zbl 1376.20053
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