Geodesic
Some conditions for embeddability of an $FC$-group in a direct product of finite groups and a torsionfree Abelian group
Sbornik. Mathematics, Tome 42 (1982) no. 4, pp. 499-514 
L. A. Kurdachenko. Some conditions for embeddability of an $FC$-group in a direct product of finite groups and a torsionfree Abelian group. Sbornik. Mathematics, Tome 42 (1982) no. 4, pp. 499-514. http://geodesic.mathdoc.fr/item/SM_1982_42_4_a3/
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     title = {Some conditions for embeddability of an $FC$-group in a~direct product of finite groups and a~torsionfree {Abelian} group},
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Voir la notice de l'article provenant de la source Math-Net.Ru

By definition, a torsionfree Abelian group $A$ belongs to the class $A(SD\mathfrak F)$ if every $FC$-group $G$ with $t(G)\in SD\mathfrak F$ and $G/t(G)\cong A$ is embeddable in a direct product of finite groups and a torsionfree Abelian group. If $A$ is a torsionfree Abelian group of rank 1, then $\operatorname{Sp}(A)=\{q, q\text{ a prime}\mid A=A^q\}$. The fundamental result of the article is the following statement. Theorem. {\it A torsionfree Abelian group $A$ belongs to the class $A(SD\mathfrak F)$ if and only if it admits a series of pure subgroups $$ (1)=A_1\leqslant A_2\leqslant\cdots\leqslant A_n\cdots\leqslant\bigcup_{n\in\mathbf N}A_n=A $$ with the following properties}: (I) {\it the quotient $A_{n+1}/A_n$ is of rank $1,$ and the set $\operatorname{Sp}(A_{n+1}/A_n)$ is finite$,$ $n\in\mathbf N;$} (II) {\it for every prime $q$, there exists a number $l(q)$ such that $q\in\operatorname{Sp}(A_{n+1}/A_n)$ whenever $n\geqslant l(q)$.} Bibliography: 9 titles.

[1] Gorchakov Yu. M., Gruppy s konechnymi klassami sopryazhennykh elementov, Nauka, M., 1978 | MR | Zbl

[2] Kurdachenko L. A., “$FC$-gruppy, periodicheskaya chast kotorykh vkladyvaetsya v pryamoe proizvedenie konechnykh grupp”, Matem. zametki, 21:1 (1977), 9–20 | MR | Zbl

[3] Kurdachenko L. A., “$FC$-gruppy, periodicheskaya chast kotorykh vkladyvaetsya v pryamoe proizvedenie konechnykh grupp”, Tez. dokl. VI Vses. simp. po teorii grupp, Naukova dumka, Kiev, 1978, 34–35

[4] Maltsev A. I., “O gomomorfizmakh na konechnye gruppy”, Uch. zap. Ivanovskogo ped. in-ta, 18 (1958), 49–60

[5] Fuks L., Beskonechnye abelevy gruppy, T. 1, Mir, M., 1974

[6] Orsatti A., “Una caratterizzatione dei gruppi abeliani compatti a lucalmente compatti nella topologia naturale”, Rend. Semin mat. Univ. Padova, 39 (1967 (1968)), 219–225 | MR

[7] Kurdachenko L. A., “O stroenii FC-grupp, periodicheskaya chast kotorykh vkladyvaetsya v pryamoe proizvedenie konechnykh grupp”, Matem. zametki, 25:1 (1979), 15–26 | MR | Zbl

[8] Kurdachenko L. A., “$FC$-gruppy s ogranichennymi v sovokupnosti poryadkami elementov periodicheskoi chasti”, Sib. matem. zh., 16:6 (1975), 1205–1213 | MR | Zbl

[9] Fuks L., Beskonechnye abelevy gruppy, T. 2, Mir, M., 1977