Residually finite groups are those groups embeddable in a direct product of a family of finite groups. What happens if we consider only restricted direct product?
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$\begingroup$ Subgroups of restricted direct products have very restrictive properties: they are locally finite, and they are FC-groups (i.e., all conjugacy classes are finite, or equivalently, every element has a centralizer of finite index). This restricts to locally finite residually finite FC-groups. Whether every such group embeds into a restricted direct product of finite groups, I don't see immediately. $\endgroup$– YCorCommented Feb 14, 2020 at 13:10
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$\begingroup$ These are locally finite FC-groups (the centralizer of every element has finite index. See "FC groups whose periodic parts can be embedded in direct products of finite groups" and references there. $\endgroup$– user6976Commented Feb 14, 2020 at 13:11
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$\begingroup$ @MarkSapir it can't be true, since there are (abelian) locally finite FC-groups that are not residually finite. $\endgroup$– YCorCommented Feb 14, 2020 at 13:12
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$\begingroup$ What cant be true? $\endgroup$– user6976Commented Feb 14, 2020 at 13:14
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1$\begingroup$ @Yves: direct sum of finite abelian groups is isomorphic to a direct sum of finite cyclic groups. Kulikov gave a general criterion for an abelian $p$-group to be a subgroup of a direct sum of cyclic groups: see Theorems III.17.1 and III.18.1 in the book "Infinite Abelian Groups" by Laszlo Fuchs. This criterion implies that the torsion subgroup of the Cartesian product $\Pi_{n \in \mathbb{N}} C_{p^n}$ cannot be embedded in a direct sum of cyclic groups. $\endgroup$– Ashot MinasyanCommented Feb 14, 2020 at 22:16
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