Some conditions for embeddability of an $FC$-group in a~direct product of finite groups and a~torsionfree Abelian group

L. A. Kurdachenko

Sbornik. Mathematics, Tome 42 (1982) no. 4, pp. 499-514

Résumé

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.