GROUPS WITH THE INFINITE NON-QUASICENTRAL NODAL SUBGROUP GROUPS WITH THE INFINITE NON-QUASICENTRAL NODAL SUBGROUP

We present some properties of the groups with the infinite non-quasicentral periodic nodal subgroup. Our main results are formulated in Theorem 1, 2 and Theorem 3, 4. 2000 Mathematics Subject Classification: 20F22, 20F25

Let R 2 =<e>, then R=R 1 is a quasicyclic group and moreover A/R,B where B is a finite Dedekind group. Thus A is the extension of the quasicyclic subgroup by the finite Dedekind group. Y.

Theorem 1.
If G is a locally graded IAN group with a nodal subgroup A, then subgroup A belongs to one of the types: 1

. A is a finite subgroup of G; 2. A is an extension of the quasicyclic subgroup by a finite Dedekind group where G´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. If A is not a quasicentral subgroup of G, then, based on Lemma 2, the subgroup A belongs to one of types 1 or 2 of this theorem. If A is a quasicentral subgroup of G, then by Lemma 1 the subgroup A belongs to one of the types 3 or 4 of this theorem. Y.
By Theorem 1 and according to the definition of 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: 1

Lemma 1.
Let G be the IAN group with a nodal subgroup A. If a nodal subgroup A contains the element of the infinite order, then A is the abelian quasicentral subgroup of group G.
Proof. If the group A contains the element x of the infinite order, then according to Definition 2, the group A is the INH group. According to Proposition 1 A is the abelian group. Let B be an arbitrary subgroup of the group A. We shall show that BCG. If B is an infinite subgroup of G, B is admittedly a normal subgroup of G.
Let B be a finite subgroup of G. If A is the abelian group containing the element x of the infinite order, then B<x>,B ×<x>. Pursuant to Definition 1 (B×<x>)CG which implies BCG. Thus A is the abelian quasicentral subgroup of the group G. Y.

If G is the locally graded IAN group with the nodal subgroup A, there exists a subgroup of A that is not a normal subgroup of G. Then A is a finite group or A is the extension of the quasicyclic subgroup by the finite Dedekind group.
Proof. Let G be the IAN group with a nodal subgroup A and let A 1 <A where A 1 is not a normal subgroup of G. Admittedly, A 1 ≠<e> is a finite subgroup of G. In agreement with Lemma 1 A is a periodic group. If A is a finite group, then this lemma is valid. Let A be an infinite periodic subgroup of G. We consider two possible cases; Case 2. Let A be ч-group. Then put A=R .B where R is the direct product of the finite number of the quasicyclic groups, R is at the same time a divisible group, and B is the finite group where B≠<e>.Therefore, A 1 is not a normal subgroup of G; there exists a cyclic subgroup <a> of A 1 that is not normal in G and where R ∩<a> =<t>. Since R is a divisible group, there exists a quasicyclic subgroup R 1 of R and furthermore R 1 contains the subgroup <t>.Put R=R 1 ×R 2 where R 2 is an infinite subgroup of A or Let G/A be a solvable INH group. By the condition 3 of Lemma 3 G/A A is the Dedekind group. Based on this fact A is an almost quasicyclic group and by Definition 1 A is a nonquasicentral of G. Suppose there exists a finite subgroup and A is normal in G. Therefore G/A is the Dedekind group, A is an almost quasicyclic group, thus G´ is a subgroup of that almost quasicyclic subgroup A. G´ of G´. Let G´ be a finite group and put A=G´. Hence A satisfies either condition 1 or condition 2 of this theorem.
If G´ is an infinite group, then |A. G´: G´|< 3, |A:A G´|< 3, too. Admittedly, A satisfies the 2 nd condition of this theorem.
Conversely. Suppose the nodal subgroup A satisfies either condition 1 or condition 2 of this theorem, then G is the IAN group with the non-quasicentral nodal subgroup A. Y By Theorem 2 and the definition of IANA groups the next corollary follows.

Let G be a locally graded IANA group with a nodal subgroup A. The nodal subgroup A of G is a non-quasicentral of G if and only if it satisfies one of the following conditions: 1. A is the finite abelian non-quasicentral subgroup of G, the quotient group G/A is the INH group with the abelian commutator or G/A is the abelian group, 2. A = Z(p∞)×D where D is the finite abelian subgroup of G, A
contains finite subgroups that are not normal in G, and G/A is the Dedekind group.

Lemma 6.
Let G be the IAN group with the infinite non-quasicentral Dedekind nodal subgroup A. Then A=Z(p∞)×D where D is a finite Dedekind group, p | |D|, and there exists the element a!A so that the subgroup <a> is not normal p -subgroup of G. For p = 2 is D´= < e>.
Proof. Let G be the IAN group with the infinite nonquasicentral Dedekind nodal subgroup A. By Lemma 1 A is an almost quasicyclic group. Pursuant to Proposition 2 A=Z(p∞)×D where D is a finite Dedekind group, p=2, and D´=<e>. We shall prove that A does not contain a normal p-group <a> of G.
Obviously, if every cyclic subgroup of A is a normal subgroup of G, then A is a quasicentral subgroup of G. Thus there exists a cyclic subgroup < x > of A that is not normal in G, which implies that the <a> Sylow q-subgroup of the group <x> is not normal in G. This verifies that q = p.
Let q ≠ p and Z(p∞)∩<a>= <e>. Then Z(p∞).<a>,Z(p∞)×<a> where Z(p∞)×<a> is the normal subgroup of G . Evidently, <a> is the Sylow q-subgroup normal in Z(p∞)×<a> and <a> is normal in G. This is a contradiction, thus q = p.
If A is a quasicentral subgroup of group G, then, analogous to the paragraph above, we can prove that G/A is the Dedekind group. Admittedly, G/A is the abelian or the Hamiltonian group. If G/A is the abelian group, then A satisfies the 1 st condition of this lemma.
Let G/A be the Hamiltonian group. By Proposition 2 G/A is a locally finite group. Thus an extension of a locally finite group by a locally finite group is a locally finite group, which implies that G is a locally finite group and hence A satisfies the 2 nd condition of this lemma. Y.

If G is the locally graded IAN group with the infinite nodal subgroup A non-quasicentral of G , then A is the extension of a quasicyclic group by the Dedekind group.
Proof. Let G be the locally graded IAN group with the infinite nodal subgroup A non-quasicentral of G. According to Theorem 1 A is a periodic group, by Lemma 3 A is the extension of a quasicyclic group by the Dedekind group. Y

If G is the group with a finite nodal subgroup A, then G/A is the abelian group, or the group.
Proof. If G/A is the abelian group, then this lemma is valid. Let G/A be a non-abelian group and B/A be an arbitrary infinite subgroup of G/A. There evidently exists BCG and furthermore

Let G be the locally graded IAN group with a nodal subgroup A. The nodal subgroup A of G is a non-quasicentral of G if and only if it satisfies one of the following conditions: 1. A is a finite non-quasicentral subgroup of G, the quotient group G/A is the INH group with the abelian commutator or G/A is the abelian group; 2. A is an almost quasicyclic group which contains the finite subgroups that are not normal in G, |A: A∩G |<3 , and G/A is the Dedekind group.
Proof. Let G be the locally graded IAN group with the infinite nodal subgroup A. Admittedly, the subgroup A is a nonquasicentral of G. Referring to Lemma 2 A is a finite group, or A is an almost quasicyclic group.
If A is a finite group then, by Lemma 5, G/A is the abelian group or G/A is a solvable INH group. According to Proposition 1 the commutator of a solvable INH group is the abelian group. Thus the subgroup A satisfies the 1st condition of this theorem.

and A/Z(p∞) is a quasicentral in G/Z(p∞).
Proof. Let G be the locally graded IAN group with the infinite non-quasicentral nodal subgroup A of G. By Theorem 1 A is an almost quasicyclic group containing the finite subgroups that are not normal in G, |A: A∩G |<∞, and G/A is the Dedekind group. The above mentioned implies that A contains a subgroup Z(p∞) that is normal in G, A/Z(p∞) is the finite Dedekind group and furthermore Z(p∞)≤B≤A. Pursuant to Definition 1 the group B is the infinite subgroup of G. Admittedly, B is normal in G and the factor group A/Z(p∞) is the quasicentral subgroup of G/Z(p∞). According to Theorem 3.1 [8] the subgroup A satisfies the conditions of this theorem. Evidently, A is the group of one of types 1 to 8 of this theorem.
If A is a group of the type 1 of Theorem 3.1 [8], then A is the Dedekind group, A = Z(p∞)×D where D is the finite Dedekind group, p = 2, and D´=<e>. By Lemma 6 p | |D|, the subgroup A contains element a so that a subgroup <a> is not normal p-subgroup in G . Thus A is of the type 1 of this theorem.
If A is a group of one of the types 2 -8 of Theorem 3.1 [8], then A is a subgroup of one of the types 2 -8 of this theorem .
Conversely. If G is a group with the normal subgroup A of one of the types 1 -8 of this theorem , then G/A is the Dedekind group. G is evidently the locally graded group. Because G/A is the Dedekind group and A /Z(p∞) is the quasicentral subgroup of G/Z(p∞), then any infinite subgroup contained in A and any subgroup which contains a subgroup A is normal in G. Thus G is the IAN group.
Let A be an infinite subgroup of G. If the subgroup A is of the type 1 of this theorem, then the subgroup A contains a subgroup <a> that is not normal in G. Thus the subgroup A is nonquasicentral subgroup of G.
Thus the quasicentral subgroups of the group G are the Dedekind groups, which implies A is a group of one of the types 2 -8 of this theorem. Thus A is the non-Dedekind group, which implies that the subgroup A of one of the types 2 -8 is a nonquasicentral subgroup of G. Y normal in G and furthermore Z( p 3 )∩<a>1<a>, |b |>1, < b> is a p-group, therefore p | |D|. Y an element a so that <a> p -subgroup is not normal in G, and A/Z (p∞) is the quasicentral in G/Z (p∞).
Let G be a group, A is a subgroup A of G, and A= Z(p∞)×D, where D is a finite group, p | |D |, the subgroup A contains an element a so that <a> p -subgroup is not normal in G, and A/Z(p∞) is the quasicentral in G/Z(p∞).
Conversely. Suppose that A C G where A is an almost quasicyclic group. Since A/Z(p∞) is a quasicentral in G/Z(p∞), then B/Z(p∞)C G/Z (p∞) for all B/Z(p∞)<A/Z(p∞). Hence BCG, A is the abelian subgroup, every infinite subgroup of A and every infinite subgroup of G containing A is a normal subgroup of G. By Definition 1 the group G is the IANA group. Hence the subgroup A contains the subgroup that is not normal in G, then A is the non-quasicentral in G. Y

Theorem 4.
The group G is the IANA group with an infinite nonquasicentral nodal subgroup A, if and only if A=Z(p∞)×D, where D is a finite group, p | |D |, the subgroup A contains an element a so that <a> p-subgroup is not normal of G, and A/Z(p∞) is the quasicentral in G/Z(p∞).
Proof. Let G be the locally graded IAN group with the infinite non-quasicentral nodal subgroup A of G, and A´=<e>. Because G/A is the Dedekind group, A=Z(p)×D, where D is the finite abelian group, A contains the finite subgroups that are not normal in G, and G/A is the Dedekind group. The group G is evidently the locally graded IAN group with the nodal subgroup A of the type 1 of Theorem 2, p | |D|, the subgroup A contains