Timeline for Embedding into a restricted direct product of finite groups
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 14, 2020 at 22:16 | comment | added | Ashot Minasyan | @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. | |
| Feb 14, 2020 at 16:21 | comment | added | YCor | I don't actually know whether every residually finite abelian $p$-group $A$ embeds into a direct sum of finite (abelian) groups. For instance, what about the torsion subgroup of $\prod_n C_{p^n}$? | |
| Feb 14, 2020 at 14:07 | comment | added | user6976 | For example Groups with Finite Conjugacy Classes YM Gorchakov - 1978 - Nauka, Moscow | |
| Feb 14, 2020 at 13:57 | comment | added | user6976 | The reference is to a paper where a description is given and papers where the theory of FC groups is developed. There are books about the subjecr. Most of results are about connections between FC -groups and direct products of finite groups. | |
| Feb 14, 2020 at 13:44 | comment | added | YCor | @MarkSapir your comment looks like the assertion that groups that are (E) embeddable into restricted direct products of finite groups are exacty those locally finite FC-groups. Possibly you were just saying that groups (E) are locally finite FC-groups, in which case it's a trivial remark which doesn't need a reference. | |
| Feb 14, 2020 at 13:14 | comment | added | user6976 | What cant be true? | |
| Feb 14, 2020 at 13:12 | comment | added | YCor | @MarkSapir it can't be true, since there are (abelian) locally finite FC-groups that are not residually finite. | |
| Feb 14, 2020 at 13:11 | comment | added | user6976 | 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. | |
| Feb 14, 2020 at 13:10 | comment | added | YCor | 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. | |
| Feb 14, 2020 at 12:09 | history | asked | Sh.M1972 | CC BY-SA 4.0 |