Infinite soluble groups.

*(Russian. English summary)*Zbl 0063.06802From 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.

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 |