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Infinite soluble groups. (Russian. English summary) Zbl 0063.06802
From the author’s summary: In the present paper the following definition of a soluble group is given: A (finite or infinite) group $$\Gamma$$ is called soluble if there exist soluble sets in all the quotient groups $$\Phi/\Psi$$, where $$\Phi$$ is an arbitrary subgroup of $$\Gamma$$ (in particular, $$\Gamma$$ itself) and $$\Psi$$ is an arbitrary normal subgroup of $$\Phi$$ (in particular, the unit of the group).
The importance of the “local” properties of groups is further established. A group is said to be locally soluble if all its subgroup generated by finite systems of elements are soluble in the above sense. We have the fundamental
Theorem 3. A locally soluble group is soluble.
We further have: Theorem 4. If a group $$\Gamma$$ is soluble, then for every normal subgroup $$\Theta$$ a soluble set of $$\Gamma$$ can be chosen, which contains $$\Theta$$ as an element.
Theorem 5. If a group possesses a well-ordered ascending soluble set, then the group is soluble.
A group is called locally finite if every subgroup generated by a finite set of elements is finite. For locally finite groups the property to be soluble is equivalent to the existence of a soluble set. The theory of such groups has been given by S. Chernikov (S. Tchernikov) [Mat. Sb., N. Ser. 13(55), 317–333 (1943; Zbl 0063.07318)].
We introduce the notion of weakly soluble set. The difference between a soluble set mentioned above and weakly soluble set of a group $$\Gamma$$ is that the elements of the latter are subgroups, not necessarily normal in $$\Gamma$$; the second condition is replaced by the following one: whenever there are two immediately neighbouring subgroups belonging to the weakly soluble set, the smaller subgroup is normal in the larger one and the corresponding quotient group is abelian.
If a locally finite group possesses at least one weakly soluble set, then the group is soluble (Chernikov, loc. cit.).
Theorem 6. If a normal subgroup $$\Theta$$ of a group $$\Gamma$$ and the quotient group $$\Gamma/\Theta$$ are locally finite, then $$\Gamma$$ itself is locally finite.
Several classes of groups that had been defined before, such as nilpotent groups, locally finite $$p$$-groups, periodical special groups [O. Schmidt (O. Yu. Shmidt), Rec. Math. Moscou, n. Ser. 8, 363–375 (1940; Zbl 0024.25403)], proved to be particular cases of soluble groups.
It is our theory that gives simple proofs of Chernikov’s theorems on locally soluble groups with the minimality condition [S. Chernikov , loc. cit.]; these theorems are thereby generalized.
Some subclasses can be distinguished in the class of soluble groups in the sense of ours that now will be called A-soluble. By a B-soluble group we shall mean a soluble group, which possesses at least one well-ordered ascending soluble set, by a C-soluble group – that possessing at least one well-ordered descending soluble set. The property B is, evidently, a generalization of the property of finite soluble group that they possess abelian normal subgroups different from the unit, while C is a generalization of another property of these groups – to be different from their commutants.
The notions of solubility introduced by R. Baer [Trans. Am. Math. Soc. 47, 393–434 (1940; Zbl 0023.30002)] are various particular cases of our B-solubility.
Finally, we give examples of groups that are B-soluble, but not C-soluble, and conversely. The first example is, at the same time, an example of infinite $$p$$-group, which coincides with its commutant, while the second one shows an infinite $$p$$-group without abelian normal subgroups.

##### MSC:
 20F16 Solvable groups, supersolvable groups
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