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Nonperiodic FC-groups and with them related classes of locally normal groups and of torsionfree Abelian groups. (Russian) Zbl 0597.20031

Let T be an SDF-group (i.e. a subgroup of a direct product of finite groups). The torsion-free abelian group A is said to be in the class A(T) if every FC-group G with torsion subgroup t(G)\(\cong T\) and G/t(G)\(\cong A\) can be embedded in a direct product of finite groups and a torsion- free abelian group. The author has earlier described the class A(T) when Z(T) has finite exponent [Ukr. Mat. Zh. 35, 374-378 (1983; Zbl 0534.20026)]. Here A(T) is determined when Z(T) has infinite exponent. If \(\pi =\pi (Z(T))\) is finite then \(A\in A(T)\) if and only if A is countable and \(A/A^ q\) is finite for all \(q\in \pi\). If \(\pi\) is infinite then \(A\in A(T)\) if and only if \(A=\cup^{\infty}_{n=0}A_ n\) is the union of a series of pure subgroups, \(1=A_ 0\leq A_ 1\leq...\), such that \(A_{n+1}/A_ n\) has rank one, \(\pi \cap Sp(A_{n+1}/A_ n)\) is finite and, for each \(q\in \pi\), there is an integer \(\ell (q)\) such that \(q\in Sp(A_{n+1}/A_ n)\) for all \(n\geq \ell (q)\). [If B is a torsion-free abelian group of finite rank then B contains a free abelian subgroup F such that B/F is torsion. If D/F is the divisible part of B/F, then the spectrum of B is \(Sp(B)=\pi (D/F).]\)
Reviewer: M.J.Tomkinson

MSC:

20F24 FC-groups and their generalizations
20E25 Local properties of groups
20E07 Subgroup theorems; subgroup growth
20E15 Chains and lattices of subgroups, subnormal subgroups
20E34 General structure theorems for groups

Citations:

Zbl 0534.20026
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