ON ONE CLASS OF THE INFINITE NON-ABELIAN GROUPS ON ONE CLASS OF THE INFINITE NON-ABELIAN GROUPS

We present some properties of one class of the infinite non-abelian groups. We deal with a generalization of the INH and KI groups. Our main results are Theorem 1, 2, 3 and Theorem 4.

The description of groups defined by the systems of their subgroups was first described in the papers by Chernikov, S. N. and Kurosh, A. G (RN -groups, [5]) Chernikov dealt with an extension of the direct product of the finite number of the quasicyclic groups by the finite abelian group (ч -groups, [1]), and with the infinite non-abelian groups whose arbitrary infinite subgroup is the normal subgroup of the whole group (INH -groups, [1]). Subbotin, I. J. studied the groups G whose every subgroup of the commutator GЈ is a normal subgroup of G (KI -groups, [6]). In this paper we describe the infinite non-abelian groups with the finite nodal subgroup.
We use the standard designations of terminology of the theory groups.
Hamiltonian group is called a non-abelian Dedekind group. A group G is called a locally graded group if every finitely generated non-trivial subgroup of G contains a proper subgroup of finite index ( [1] p. 236). The subgroup A is called quasicentral of the group G if each subgroup of A is the normal subgroup of G. If ! / a group G is an extension of the group H by the normal subgroup N of G, then G/N Х H. If a group G is an extension of the quasicyclic group by the finite group, then a group G is called an almost quasicyclic group. Definition 1. An infinite non-abelian G is said to be an IAN group if there exists a subgroup A of G such that every infinite subgroup of A and every infinite subgroup of G contains A is a normal subgroup of G. The subgroup A is called a nodal subgroup. If A is an abelian subgroup, then G is IANA group. Let F be a group and let X ϭ {x i ; i ʦ I} Ó ր be a subset of F such that Ͻ x i Ͼ an infinite cyclic group for all x i from X. If F ϭ Ͻx i Ͼ is the direct sum of the cyclic groups Ͻ x i Ͼ for every i ʦ I, then F is free abelian group with the set X of the free generators.

Lemma 1.
If G is IAN group with a finite nodal subgroup A, then G/A is an abelian group or INH group.

Proof.
If put A ϭ GЈ, evidently, then G is IAN group with a finite nodal subgroup.

Lemma 3.
Let G be IAN group with a nodal subgroup A. If a nodal subgroup A contains the elements of the infinite orders, then A is an abelian quasicentral subgroup of group G.
Proof. By Proposition 2 A is an abelian group. Let B be an arbitrary subgroup of the group A. If B is an infinite subgroup, then B is a normal subgroup of G.
Let B be a finite subgroup. By assumption, the group A contains the element x of the infinite order, A is an abelian group.
Thus A is an abelian quasicentral subgroup of group G. Lemma is proved.

Lemma 4.
Let G be a locally graded IAN group with a nodal subgroup A. If there exists a subgroup A that is not a normal subgroup of G, then A is a finite group, or A is an extension of the quasicyclic subgroup by a finite Dedekind group.
If A is a finite group, then lemma is valid. Let A be an infinite periodic subgroup. We consider two possible cases: A is non ч -group, or A is чgroup.
Let A be non ч -group. Suppose that there exists the sub- Let A be ч -group. Denote A ϭ R.B, R is the direct product of the finite number of the quasicyclic groups, R is a divisible group, B is the finite group, B Ͻ e Ͼ. Therefore, A 1 is not a nonnormal subgroup of G. we assume that there exists a cyclic subgroup Ͻ a Ͼ of A 1 non-normal in G, and also R പ Ͻ a Ͼ ϭ Ͻ t Ͼ.
Since R is a divisible group there exists a quasicyclic subgroup R 1 of R and, furthermore, It is contradiction. Then R 2 ϭ Ͻ e Ͼ, R ϭ R 1 is quasicyclic group, and moreover, A/R Х B is a finite Dedekind group. Thus A is an extension of the quasicyclic subgroup by the finite Dedekind group. Lemma is proved.

Theorem 1.
If G is a locally graded IAN group with a nodal subgroup A, then subgroup A belongs to one of the types: A is an extension of the quasicyclic subgroup by a finite Dedekind group, the commutator GЈ is an infinite group.

3.
A is an infinite quasicentral periodic subgroup of G, 4. A is a quasicentral non-periodic abelian subgroup of G.
Proof. Let A be a non-quasicentral subgroup of G. By Lemma 4 the subgroup A of G belongs to one of type 1 or 2 of this theorem.
Let A be a quasicentral subgroup of G. By Lemma 3 the subgroup A of G belongs to one of type 3 or 4 of this theorem. Theorem is proved.
By Theorem 1 and definition IANA groups the next corollary follows.

Corollary 1.
If G is a locally graded IANA group with a nodal subgroup A, then subgroup A belongs to one of the types: Proof. Let G be IAN group with a finite nodal subgroup A. If G/A is Dedekind group, then G is of type 1. If G/A is non-Dedekind group, by Lemma 1 G/A is non-Hamiltonian INH group. If G/A is a non-solvable group, then G is of type 2.

If G/A is a solvable group by Proposition 1 G/A ϭ (D/A).(H/A) where D/A A G/A, D/A is a quasicyclic group, H/A is such a finite subgroup of G that the factor group H/A/(D/A പ H/A) is Dedekind group. Denote G ϭ D.H, where D A G, H is a finite group, and
A Յ D പ H. If GЈ is a finite group, then G/GЈ is an infinite group. By Lemma 2 G is a group of type 1. Conversely, let G be the group belonging to one of types 1 to 3. We shall prove that G is group with a finite nodal subgroup. Let the group G be of type 1. Then G is an extension of the finite sub-group N by an infinite Dedekind group. Thus the commutator of the group G/N is a finite group, evidently the commutator of the group G is a finite group too. By Lemma 2 G is group with a finite nodal subgroup.

Let GЈ be an infinite group. Assuming that D/A is a quasicyclic group, A is a finite group. Denote
Let G be the group of type 2 or 3. Suppose that there exists a finite normal subgroup N of the group G so that G/N is non Dedekind group. Put N ϭ A. Let B be an arbitrary infinite subgroup of G containing A. Then B/A A G/A implies B A G. Since A is a finite group, evidently its every infinite subgroup is normal of G. Thus the groups of type 2 or 3 are IAN groups with a finite nodal subgroup. Theorem is proved.

Corollary 2.
The locally graded IAN groups with a finite nodal subgroup are the groups to one of type 1 or 3 of Theorem 2.
Proof. If G is an locally graded IAN group with a finite nodal subgroup A, then G is the group to one of type 1 to 3 of Theorem 2. If G is the group to one of types 1 or 3 of Theorem 2, then Corollary 2 is valid. Prove that G is not the group of type 2 of Theorem 2. Let G be the group of type 2 of Theorem 2, then G/A is an infinite non-solvable INH group. If A is a finite group, and G locally graded group, evidently then G/A is a locally graded nonsolvable INH group. It is contradiction. Thus G is not the group of type 2 of Theorem 2.

Corollary 3.
The locally graded IAN group with a finite solvable nodal subgroup is the solvable group belonging to one of types 1 or 3 of Theorem 2.
Proof. Let G be an investigated group, A is its finite solvable subgroup. By Corollary 2 G/A is the group to one of types 1 or 3 of Theorem 2. Evidently, then G/A is the solvable group, by assumption A is the solvable subgroup. This implies that G is the solvable group. Theorem 3. The locally graded IANA group with a finite nodal subgroup is the solvable group of degree less or equal to number 3.
Proof. If G/A and A are solvable groups, then G is the solvable group and (G/A)Ј is abelian group. Using this fact it follows that G (2) is abelian group and G (3) ϭ Ͻ e Ͼ. Theorem is proved. Theorem 4. The locally graded IANA groups with a finite nodal subgroup are the groups belonging to one of the types: 1. G is an extension of the finite normal abelian subgroup N by Dedekind group. 2. G is an almost quasicyclic group, GЈ is an almost quasicyclic abelian group. 3. G ϭ R.H, R is a p-quasicyclic group, R A G, H is a finite group containing a normal subgroup N of G so that H/N is Hamiltonian group, R.N is abelian group, the commutator HЈ is nonabelian group.
Proof. Let G be locally graded IANA groups with a finite nodal subgroup. By Theorem 3 G is the solvable group. If G/A is abelian group, then G is the group of type 1 of this theorem.
Let G/A be non-abelian group. By Lemma 1 G/A is INH group. By Proposition 1 G/A is Hamiltonian group or an extension of the quasicyclic subgroup by the finite Dedekind group. If G/A is Hamiltonian group, then G is the group of type 1 of this theorem.
Let G/A be an extension of the quasicyclic subgroup B/A by the finite Dedekind group. We know that A is a finite abelian group, thus |B : C B (A)| Ͻ ϱ and A Յ C B (A). If A Յ C B (A) and B/A is a quasicyclic group which does not contain the proper subgroup of the finite index, then C B (A) ϭ B and thus A Յ C B . It follows that B is a central extension of the finite abelian group A by the quasicyclic group (B is an almost quasicyclic group). Since R is a quasicyclic subgroup of B and a characteristic subgroup of B, thus R is a normal subgroup of G. By Isomorphism Theorem (G/A)/(B/A) Х G/B and furthermore G/B is finite Dedekind group. We consider two possible cases, G/B is an abelian group or Hamiltonian group.
Let G/B be an abelian group. Evidently, the commutator GЈ Յ B. If GЈ is a finite group, then G is the group of type 1 of this theorem.
If GЈ is an infinite group, evidently GЈ Յ B, then G and GЈ are almost quasicyclic groups. Thus G is the group of type 2 of this theorem. If GЈ is a finite group, then G is the group of type 1 of this theorem. If GЈ is an infinite group, then G is the group of type 2 of this theorem.
Let GЈ be an infinite non-abelian group. Put G ϭ R.H, where R is a normal subgroup of G. It is well-know that C G (R) Ն G, GЈ ϭ RЈ.HЈ . [R,H] Յ R.HЈ, and [R.HЈ] ϭ Ͻ e Ͼ. From the product of the groups M and N, where M is the central group and N is abelian group and GЈ Յ R.HЈ, it follows that HЈ is nonabelian group. Thus G is the group of type 3 of the Theorem 4.
Conversely, let G be an infinite group one of types 1 to 3. If G is one of type 1 or 3, then we put A ϭ N. If G is of type 2 we put GЈ ϭ R ϫ D which is the direct product of a quasicyclic group R and a finite group D. If A is the subgroup of G generated by the elements of GЈ whose orders are divisors of the order of D, then A is abelian group and D Յ A.
Let G be the group of type 1. Then G/A is Dedekind, there exits B/A an arbitrary normal subgroup of G/A. Consequently, B A G.
Let G be the group of type 2. We have GЈ ϭ R ϫ D ϭ R.A. It follows that G/A is an extension of the quasicyclic group GЈ/A ϭ ϭ R.A/A by the finite abelian group. Thus B/A is an arbitrary normal subgroup in G/A. Consequently, B A G.
Let G be the group of type 3. In that case G/A is an extension of the quasicyclic group R.A/A by the finite Hamiltonian group H/A. Evidently, B/A A G implies B A G. If G is the group one of types 1 to 3 this theorem, then G is IANA group with a finite nodal subgroup A. Theorem is proved.
Remark. The class of the infinite solvable IANA groups contains the class of solvable INH and the class KI groups with the finite commutator.