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Locally finite groups containing a finite inseparable subgroup. (English. Russian original) Zbl 0836.20051
Sib. Math. J. 34, No. 2, 218-232 (1993); translation from Sib. Mat. Zh. 34, No. 2, 23-41 (1993).
Let $$G$$ be an infinite locally finite group and let $$H_0$$ be an arbitrary finite subgroup in $$G$$. Choose a finite nontrivial $$H_0$$-invariant subgroup $$H_1 < G$$ with $$H_1 \cap H_0 = 1$$. For the subgroup $$\langle H_0, H_1\rangle$$, find again a finite nontrivial $$\langle H_0, H_1\rangle$$-invariant subgroup $$H_2$$ with $$\langle H_0, H_1\rangle \cap H_2 =1$$. Continuing this process further, we thus construct an infinite subgroup $$\langle H_0, H_1, H_2, \dots \rangle$$ which is obviously a residually finite group. Executing such a procedure in an arbitrary locally finite group, one sometimes faces the impossibility of choosing the subgroup $$H_i$$ at some step of the construction. For instance, the procedure is obviously impossible in the case of a quasi-cyclic group or, more generally, a Chernikov group. It is interesting to classify locally finite groups in which constructing infinite subgroups causes none of the difficulties indicated. To specify the question, we give the following definition. A subgroup $$H$$ of a group $$G$$ will be called separable in $$G$$, if there is a nontrivial $$H$$- invariant subgroup $$K < G$$ such that $$H \cap K = 1$$. Otherwise, we call $$H$$ inseparable. Thus, the main purpose of the present article is to investigate the structure of locally finite groups containing a finite inseparable subgroup.

##### MSC:
 20F50 Periodic groups; locally finite groups 20E07 Subgroup theorems; subgroup growth 20E25 Local properties of groups 20E26 Residual properties and generalizations; residually finite groups
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##### References:
 [1] V. V. Belyaev, ?Locally finite groups containing a finite inseparable subgroup,? Sibirsk. Mat. Zh.,34, No. 2, 23-41 (1993). · Zbl 0836.20051
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