On Torsion and Finite Extension of \(FC\) and \(\tau N_{K}\) Groups in Certain Classes of Finitely Generated Groups

1. Introduction

Definition 1.1. A group \(G\) is said to be with finite conjugacy classes (or shortly \(FC\)-group) if and only if every element of \(G\) has a finite conjugacy class in \(G\).

It is known that \(FIZ\subseteq FA\subseteq FC\), where \(FIZ\) denotes the class of center-by-finite groups, and that for finitely generated equalities \(FIZ=FA=FC\) hold. These results and other have been studied and developed by Baer, Neumann, Erdos and Tomkinson and others in [1, 2, 3, 4, 5]. \(FC\)-groups have many similar properties with abelian groups and finite groups. It is known that the class of \(FC\)-groups is closed under taking subgroup, homomorphic images, quotient and forming restricted direct products, but it is not closed under taking finite extension. We prove in Theorem 2.2, that in the class of finitely generated \(\tau N\)-group (respectively \(FN\)-group) a \((FC)\tau \)-group (respectively \((FC)F\)-group) is \(\tau A\)-group (respectively is \(FA\)-group).

Definition 1.2. Let \(\chi \) is a given property of groups. A group \(G\) it is said to be in the class \((\chi ,\) \(\infty )\) (respectively \((\chi ,\infty)^{\ast }\)) if and only if every infinite subset \(X\) of \(G\) contains two distinct elements \(x\), \(y\) such that the subgroup \(< x\), \(y >\) (respectively \(< x\), \(x^{y}>\)) is a \(\chi\)-group. Note that if \(\chi\) is a subgroup closed class, then \(\chi \subset (\chi , \infty )\ \subset (\chi , \infty)^{\ast }\).

On the one hand, several authors have studied the class of \((\chi , \infty)\)-groups, where \(\chi \) is a given property of groups, with some conditions on these groups. The question that interests mathematicians is the following: If \(G\) is a group in the class \((\chi ,\infty )\) where \(\chi\) is a given property, then does \(G\) have a property in relation to the property \(\chi\). For example \(G\) has the property \(\chi \gamma\) or \(\gamma \chi\), etc. where \(\gamma\) is another group property, or in particular \(G\) is in the same class \(\chi\). For example, in 1976, Neumann in [6], has shown that a group is in the class \((A,\infty)\) if and only if it is \(FIZ\)-group. In 1981, Lennox and Wiegold in [7] proved that a finitely generated solvable group is in the class \((N,\infty )\) (respectively \((P,\infty )\), \((C_{o},\infty )\)) if and only if it is \(FN\), (respectively \(P\), \(C_{o}\)), where \(P\) and \(C_{o}\) respectively polycyclic and coherent groups.

In 2000, 2002 and 2005, Abdollahi and Trabelsi, proved in [8, 9, 10] that a finitely generated solvable group is in the class \((FN_{k}, \infty )\) (respectively \((FN, \infty)\), \((NF, \infty )\), \((\tau N, \infty )\)) if and only if it is \(FN_{k}^{(2)}\), (respectively \(FN,NF,\tau N\)). Other results of this type have been obtained, for example in [8, 9, 10, 11, 12, 13, 14, 15, 16]. In this note, we prove that a finitely generated \(\tau N\)-group \(G\) which is in the class \(((\tau N_{k})\tau ,\infty)\) is in the class \(\tau N_{k}^{(2)}\) and deduce that a finitely generated \(FN\)-group (respectively \(NF\)-group) \(G\) in the class of \(((FN_{k})F,\infty )\)-groups, is in the class of \(FN_{k}^{(2)}\)-groups (respectively in the class of \(N_{k}^{(2)}F\)-groups) and particularly a group \(G\) is in the class \(((FC)F,\infty )\) if and only if, it is \(FA\)-group.

On the other hand, in 2005, Trabelsi in [10](respectively in 2007, Rouabehi and Trabelsi in [17]) proved that a finitely generated soluble group in the class \((CN, \infty )^{\ast }\) where \(C\) is the class of cernikov group (respectively in the class in the class \((\tau N,\infty )^{\ast }\)) is \(FN\)-group (respectively \(\tau N\)-group) and in 2007 too, Guerbi and Rouabhi in [14] proved that a finitely generated Hyper(abelian-by-finite) group in the class \((\Omega ,\infty )^{\ast }\) where \(\Omega\) is the class of finite depth group, is \(FN\)-group. In this paper, we prove that a finitely generated \(\tau N\)-group in the class \(((FN_{k})\tau ,\infty )^{\ast }\) is in the class \(\tau N_{c}\) for certain integer \(c\) and deduce that a finitely generated \(FN\)-group (respectively \(NF\)-group) \(G\) in the class \(((FN_{k})F,\infty )^{\ast }\)) is in the class \(FN_{c}\) (respectively \(N_{c}F\)). In particular, if \(G\) is a finitely generated \(FN\)-group in the class \(((FC)F, \infty )^{\ast }\) (respectively \(((FN_{2})F, \infty )^{\ast }\)) then \(G\) is in the class of \(FN_{2}\)-groups (respectively in the class of \(FN_{3}^{(2)}\)-groups).

2. Main Results

2.1. Torsion and finite extension of property \(FC\)

It is known that the property \(FC\) is not closed under the formation of extension. The following example shows that even, a finite extension (respectively torsion extension) of an \(FC\)-group is not always an \(FC\)-group(respectively \(\tau A\)-group).

Note that if the center of an infinite finitely generated group is trivial or finite then this group is not \(FC\).

Example 2.1. Let \(G=D_{\infty }= < a,b/\) \( a^2 = 1 \) and \( aba =b^{-1} > \) the infinite dihedral group, which is a finitely generated soluble group, generated by the involutions \(a\), \(b\). We have \(K=C_{\infty }=< b >\) which is a infinite cyclic group isomorphic to \(\mathbb{Z}\) therefore it is a \(FC\)-group and the quotient group \(G/K\) is isomorphic to \(C_{2}=< a > \) which is finite of order \(2\), thus \(G\) is a finite extension of a \(FC\)-group, but as the center of the infinite dihedral group is trivial then it is not a \(FC\)-group.

This example shows also that \(D_{\infty }\) is a torsion extension of a \(FC\)-group but it is not a \(\tau A\)-group, so we consider the class of finitely generated \(\tau N\)-groups (respectively \(FN\)-group) and we prove that, in this class, a \((FC)\tau \)-group (respectively \((FC)F\)-group) is a \(\tau A\)-group (respectively \(FC\)-group).

Theorem 2.2. Let \(G\) a finitely generated torsion-by-nilpotent group. If \(G\) is \(FC\)-by-torsion group then \(G\) is \(\tau A\)-group.

Lemma 2.3. If \(G\) is a nilpotent group of nipotency class \(d\) and \(g\) an element of \(G.\) The subgroup \(< G^{\prime },g>\) generated by the derived group \(G^{\prime }\) and \(G\) is a nilpotent group of class \(\leq d\).

Lemma 2.4. If \(G\) is nilpotent and torsion-free group, \(m\), \(n\) two non-zero integers and \(x\),\(y\) \(\in G\), then,

1. If \(x^{n}=y^{n}\) then \(x=y\).
2. If \([x^{m}y^{n}]=1\) in \(G\), then \([x\ y]=1\) in \(G\).
3. If \([x^{m}yx^{n}y]=1\) in \(G\), then \([x\ y]=1\) in \(G\).

Proof. (1) We proceed by induction on the nipotency class \(d\) of the group \(G.\) If \(d=1\) so \(G\) is abelian: \(x^{n}= y^{n}\Longleftrightarrow x^{n}y^{-n}=1\Longleftrightarrow (xy^{-1})^{n}=1\) and as \(G\) is torsion-free then \(xy^{-1}=1\Longleftrightarrow x=y\). We suppose now that \(G\) is torsion-free nilpotent and non abelian of nilpotecy class \(d.\) We consider the subgroup \(H=< G^{\prime } x>\) generated by the derived group \(G^{\prime }\) and the element \(x\), by the Lemma 2.3 above the nilpotency class of \(H\) is less than \(d.\) Then by the inductive hypothesis the Lemma is verified for \(H.\) we have \(x\in H\) and \(x^{y}=y^{-1}xy=x[x y]\in H\) and \(x^{n}=y^{n}.\) So as \((y^{-1}xy)^{n}=y^{-1}x^{n}y=y^{-1}y^{n}y=y^{n}=x^{n}.\) The (1) in lemma applied to \(H\) give us that \(y^{-1}xy=x\) which means that \(x\) and \(y\) commute. So we have in \(G : x^{n}=y^{n}\Longleftrightarrow x^{n}y^{-n}=1\Longleftrightarrow (xy^{-1})^{n}=1\Longleftrightarrow x=y\).
(2) We have: \(x^{m}y^{n}=y^{n}x^{m}\Longleftrightarrow y^{-n}x^{m}y^{n}=x^{m}\Longleftrightarrow(y^{-n}xy^{n})^{m}=x^{m}\Longleftrightarrow y^{-n}xy^{n}=x\) (according to 1)\( \Longleftrightarrow xy^{n}=y^{n}x\Longleftrightarrow xy^{n}x^{-1}=y^{n}\Longleftrightarrow (xyx^{-1})^{n}=y^{n}\) also by (1) we obtain \(xyx^{-1}=y\) and so \(xy=yx.\)
(3) \([x^{m}y x^{n}y]=1\Longleftrightarrow x^{m}y x^{n}y=x^{n}y x^{m}y\Longleftrightarrow x^{m-n}y x^{n}=y x^{m}\Longleftrightarrow x^{m-n}y =y x^{m-n}\Longleftrightarrow \lbrack x^{m-n} y]=1\) by the (2) we obtain \(xy=yx\).

Proof of Theorem 2.2

Proof. Since \(G\) is finitely generated torsion-by-nilpotent group, there exists a normal and torsion subgroup \(F\) of \(G\) such that the quotient group \(G/F\) is nilpotent group. As the property \(FC\)-by-torsion is closed under quotient, it is enough to show that \(G/F\) is a \(FC\)-group. For this it is sufficient to show that every \(FC\)-by-torsion group \(G\) in the class of finitely generated nilpotent groups in a \(\tau A\)-group. Assume that \(G\) is \((FC)\)-by-torsion, so there exists a normal \(FC\)-subgroup \(N\) such that the quotient \(G/N\) is torsion. Since \(G\) is finitely generated and nilpotent, it checks the maximal condition on subgroups. So \(N\) is finitely generated \(FC\)-subgroup. According to ([1], Theorem 6.2 ) \(N\) is center-by-finite which means that \(Z(N)\) is of finite index in \(N\). Or \(G/N\) is torsion group. It follows that the quotient \(G/Z(N)\) is torsion group. So for all \(x\) and \(y\) in \(G\), there exist non-zero integers \(m\) and \(n\) such that \(x^{m}\) and \(y^{n}\) belong to \(Z(N)\), it follows that \([x^{m},y^{n}]=1\) in \(G\). Let \(T=Tor(G)\), the torsion subgroup of \(G\). Since \(G\) is nilpotent, the quotient group \(G/T\) is a finitely generated nilpotent torsion-free group, and therefore \([x^{m}T,y^{n}T]=T\) in \(G/T\). By the result (2) in Lemma 2.4 above, we deduce that \([xT,yT]=T\) in \(G/T\), which shows that the group \(G/T\) is an abelian group. More, since \(G\) is nilpotent and checks the maximal condition, by ([2], Theorem 5.1) the subgroup \(T\) as finitely generated nilpotent torsion group, is finite. Thus \(G\) is finite-by-Abelian so \(G\) is \(\tau A\). Since \(G/F\) is \(FC\)-by-torsion group in the class of finitely generated nilpotent-group, then \(G/F\) is \(\tau A\) and as \(F\) is torsion group it follows that \(G\) is \(\tau (\tau A)=\tau A\). This completes the proof.

Remark 2.1. The example below shows that Theorem 2.2. is falls when the condition "finitely generated" is omitted.

Example 2.5. Let \(A=F_{2}[X]\) algebra of polynomials on the field \(F_{2}\) and the isomorphism \(\varphi :A\times A\rightarrow A\times A\), \((P,Q)\mapsto (P+Q,Q)\). We put \(H=A\times A\) and \(K=< \varphi >\) such that \(\varphi^{2}=Id_{A\times A}\) the identity application on \(A\times A\). Since \(H\) is an abelian group, it is a \(FC\)-group. \(K\) is a finite group of order 2 and so it is \(FC\) too. We consider \(G=H\rtimes K\), the semi-direct product of \(H\) by \(K\). \(G\) is a non-finitely generated nilpotent group, which is a finite extension of the \(FC\)-group \(H\). But \(G\) is not a \(FC\)-group.

If we replace the property \(\tau\) by the property \(F\) we obtain a necessary and sufficient condition for the property \(FC\) to be closed under finite extension in the class of finitely generated \(FN\)-group.

Corollary 2.6. Let \(G\) a finitely generated finite-by-nilpotent group. \(G\) is \(FC\)-by-finite group if and only if \(G\) is \(FA\)-group.

Proof. It clear that if \(G\) is \(FA\)-group then \(G\) is \(FC\) and so \(FC\)-by-finite. If \(G\) is finitely generated finite-by-nilpotent group, as the same case in theorem above there exists a normal and finite subgroup \(F\) of \(G\) such that the quotient group \(G/F\) is nilpotent group. As the property \(FC\)-by-finite is closed under quotient, it is enough to show that \(G/F\) is a \(FC\)-group. For this it is sufficient to show that every \(FC\)-by-finite group \(G\) in the class of finitely generated nilpotent groups is a \(FA\)-group. Assume that \(G\) is \((FC)\)-by-finite, so there exists a normal \(FC\)-subgroup \(N\) such that the quotient \(G/N\) is finite. As the same way in the above theorem, we found that \(Z(N)\) is of finite index in \(N\) and the quotient \(G/Z(N)\) is finite group. So for all \(x\) and \(y\) in \(G\), there exist non-zero integers \(m \) and \(n\) such that \(x^{m}\) and \(y^{n}\) belong to \(Z(N)\), it follows that \([x^{m},y^{n}]=1\) in \(G\). If \(T=Tor(G)\), we have \([x^{m}T, y^{n}T]= T\) in \(G/T\). By the result (2) in Lemma 2.3, we deduce that \([xT, yT]=T\) in \(G/T\), which shows that the group \(G/T\) is an Abelian group.
Moreover, as in the above theorem we found that \(T\) is finite. Thus \(G\) is finite-by-Abelian. Since \(G/F\) is \(FC\)-by-finite group in the class of finitely generated nilpotent-group, it is \(FA\) and so \(G\) is \(F(FA)=FA\). This completes the proof.

\(\tau N_{k}\) and \(FN_{k}\)-groups and conditions on infinite subsets

Our first elementary propositions below follows from a results in [8, 12] and [10].

Proposition 2.7. If \(G\) is a finitely generated finite-by-soluble group in the class \((FN_{k}, \infty),\) then \(G\) is in the class of \(FN_{k}^{(2)}\)-groups.

Proof. Suppose that \(G\) is finite-by-soluble, there exists finite normal subgroup \(N\) such that \(G/N\) is soluble. As the class of \((FN_{k},\infty )\)-group, is closed under taking quotient, then the quotient group \(G/N\) is a finitely generated soluble group in the class of \((FN_{k},\infty )\)-group. By ([8] Corollary 1.8), \(G/N\) is in the class of \(FN_{k}^{(2)}\)-groups. Therefore \(G\) is finite-by-\(FN_{k}^{(2)}\)-group, and this gives that \(G\) is \(FN_{k}^{(2)}\)-group.

Proposition 2.8. If \(G\) is a finitely generated torsion-by-soluble group in the class \((\tau N_{k},\infty ),\) then \(G\) is in the class of \(\tau N_{k}^{(2)}\)-groups.

Proof. Suppose that \(G\) is finite-by-soluble, there exists a torsion and normal subgroup \(N\) such that \(G/N\) is soluble. As the class of \((\tau N_{k},\) \(\infty)\)-group, is closed under taking quotient, then the quotient group \(G/N\) is a finitely generated soluble group in the class of \((\tau N_{k},\infty )\subset (\tau N,\infty )\). By ([10], Theorem 1) \(G/N\) is in the class of \(\tau N\)-groups.
So \(G/N\) admits a torsion group \(\tau (G/N)=T/N\) such that \(T\) is torsion and the quotient \(G/T\) is torsion-free in the class \((\tau N_{k},\infty )\). So \(G/T\) is a finitely generated soluble group in the class \((N_{k}, \infty).\) It results by ([12]) that \(G/T\) \(\in FN_{k}^{(2)}\), therefore \(G\) is torsion-by-\(FN_{k}^{(2)}\), and this gives that \(G\) is \(\tau N_{k}^{(2)}\)-group.

Theorem 2.9. Let \(G\) a finitely generated \(\tau N\)-group. If \(G\) is in the class \(((\tau N_{k})\tau,\) \(\infty)\), then

1. \(G\) is \(\tau N_{k}^{(2)}\)-group.
2. There exist integers \(d\) such that \(G\) is in the class \(\tau N_{k^{d-1}}\).

Proof. (1) Assume that \(G\) is finitely generated \(\tau N\)-group in the class \(((\tau N_{k})\tau ,\infty)\). There exist a normal and torsion subgroup $H$ of \(G\) such that \(G/H\) is nilpotent quotient group. Since \(G/H\) is finitely generated nilpotent group, it has a torsion subgroup \(T/H\) of finite order and as \(H\) is torsion group then \(T\) is torsion group too. So \(G/T\) is torsion-free nilpotent group in the class \(((\tau N_{k})\tau ,\infty)\), which gives that \(G/T\) is in the class \((N_{k}\tau ,\infty)\). We deduce by ([18], Lemma 6.33) that \(G/T\) is in the class \((N_{k},\infty)\) and so \(G/T\) is a finitely generated soluble group in the class \((N_{k},\infty)\). It follows by [8] that \(G/T\) belongs in the class of \(FN_{k}^{(2)}\)-groups and torsion-free so \(G/T\) is in the class \(N_{k}^{(2)}\), it gives that \(G\) is in the class of \(\tau N_{k}^{(2)}\)-groups.
(2) In (1) we have \(G/T\) is a torsion-free nilpotent group in the class \(N_{k}^{(2)}\) which is included in \(\varepsilon _{k}\), so \(G/T\) is \(k\)-Engel torsion-free nilpotent (so soluble) group. If the integer \(d\) is the derived4length of \(G/T\) as a soluble group, then by a result of Gruenberg [18], Theorem 7.36, \(G/T\) is in the class \(N_{k^{d-1}}.\) So as \(T\) is torsion. It gives that \(G\) is \(\tau N_{k^{d-1}}\). This completes the proof.

If we replace the property \(\tau N\) by the property \(FN\), we obtain the results in the Lemma bellow.

Lemma 2.10. Let \(G\) a finitely generated \(FN\)-group in the class \(((FN_{k})F,\infty)\), then,

1. \(G\) is in the class of \(FN_{k}^{(2)}\)-groups.
2. There exist integers \(d =d(k)\) and \(c=c(k, d)\) such that \(G\) is in the class \(FN_{k^{d-1}}\) and \(G/Z_{c}(G)\) is finite.

Proof. (1) Assume that \(G\) is finitely generated \(FN\)-group in the class \(((FN_{k})F,\infty)\subset ((\tau N_{k})\tau ,\infty)\). As \(G\) is \(FN\)-group, there exist a normal and finite subgroup \(H\) of \(G\) such that \(G/H\) is nilpotent. We found that the torsion subgroup \(T/H\) of \(G/H\) is finite and so \(T\) is finite too. As the property \(((\tau N_{k})\tau,\infty)\) is closed under quotient then the quotient group \(G/T\) verifies the conditions of the above theorem. It follows that \(G/T\) belongs in the class of \(\tau N_{k}^{(2)}\)-groups which gives that \(G/T\) is in the class of \(N_{k}^{(2)}\)-groups and so G is \(FN_{k}^{(2)}\).
(2) In one hand as the same way in (2) of the above theorem we found that \(G/T\) is in the class \(N_{k^{d-1}}\) and \(T\) is finite. So \(G\) is in the class \(FN_{k^{d-1}}\). In the other hand and by Hall [15], there exist an integer \(c=c(k, d)\) depending on \(k, d\) such that \(G/Z_{c}(G)\).

The example 2.5 above shows that nilpotency is necessary for the results of the above theorem to remain true. Recall that \(FN\)-groups are \(NF\)-groups (see [15]).

Theorem 2.11. Let \(G\) a finitely generated \(NF\)-group. If \(G\) in the class \(((FN_{k})F, \infty)\), then

1. \(G\) is in the class of \(N_{k}^{(2)}F\)-groups. In particular, if \(G\) in the class \(((FA)F, \infty )\), then \(G\) is in the class of \(AF\)-groups.
2. There exist integers \(d=d(k)\) such that G is in the class \(N_{k^{d-1}}F\).

Proof. (1) Assume that \(G\) is an infinite finitely generated \(NF\)- group in the class \(((FN_{k})\tau , \infty )\). As the group \(G\) is \(NF\)-group, and then it contains a normal nilpotent subgroup \(N\) such that \(G/N\) is finite. As the subgroup \(N\) is finitely generated and nilpotent of finite index then \(N\) is polycyclic so by ([19], Theorem 5.4.15) there exist a subgroup \(M\) normal in \(N\) and poly-infinite cyclic, hence torsion-free and of finite index in \(N\). Let \(K=M_{G}\) the core of the subgroup \(M\), so \(K\) is nilpotent torsion-free of finite index in \(G\). Since the class \(((FN_{k})\tau,\infty)\) is closed under taking subgroups, then \(K\) is nilpotent subgroup in the class \(((FN_{k})\tau,\infty)\) and according to (1) of lemma 2.10, we deduce that \(K\) is torsion-free subgroup in the class of \(\tau N_{k}^{(2)}\)-groups which gives that \(K\) is \(N_{k}^{(2)}\)-group and so \(G\) is \(N_{k}^{(2)}F\)-group. In particular, for \(k=1\) we have: \(FN_{1})\tau =(FC)\tau=(FA)\tau\) and \(N_{1}^{(2)}\tau =A\tau\).
(2) As \(K\) is a torsion-free nilpotent subgroup in the class \(N_{k}^{(2)}\) then it is in the class \(\varepsilon_{k}\) of k-Engel groups. So by Gruenberg [[18], Theorem 7.36(1)], there exist integer \(d=d(k)\) such that \(K\) is in the class \(N_{k^{d-1}}\) and as \(K\) is of finite index then \(G\) is in the class of \(N_{k^{d-1}}F.\) This completes the proof.

In 2007 T.Rouabehi and N.Trabelsi in [17] proved that a finitely generated soluble group in the class \((\tau N_{k},\infty )^{\ast }\) is in the class \(\tau N_{c}\) for certain integer \(c\) depending only on \(k.\)
If we replace the properties \(((\tau N_{k})\tau , \infty )\) and \(((FN_{k})F,\infty )\) in the above results by the properties \(((\tau N_{k})\tau,\infty )^{\ast}\) and \(((FN_{k})F,\infty)^{\ast}\), we obtain the next results.

Theorem 2.12. Let \(G\) a finitely generated \(\tau N\)-group. \(G\) is in the class \(((\tau N_{k})\tau, \infty)^{\ast}\), then there exist an integer \(c=c(k)\) such that \(G\) is in the class of \(\tau N_{c}\)-group.

Proof. Assume that \(G\) is finitely generated \(\tau N\)- group in the class \(((\tau N_{k})\tau,\infty )^{\ast}\). There exist a normal and torsion subgroup \(F\) of \(G\) such that \(G/F\) is nilpotent quotient group. Since \(G/F\) is finitely generated nilpotent group, it has a finite and so torsion subgroup \(T/F\) such that \(T\) is a normal and torsion subgroup containing \(F\). So \(G/T\) is torsion-free nilpotent group in the class \(((\tau N_{k})\tau,\infty)^{\ast}\) and hence \(G/T\) is in the class \((N_{k}\tau,\infty)^{\ast }\). We deduce by ([18], Lemma 6.33) that \(G/T\) is in the class \((N_{k},\infty)^{\ast}\). It is known that the class \((N_{k},\infty )^{\ast }\) is included in the class \(\varepsilon _{k+1}(\infty)\), where \(\varepsilon_{k+1}(\infty)\) is the class of groups whose every infinite subset \(X\) contain two distinct elements \(x,\) \(y\) such that \([x,_{k+1}y]=1\). We deduce that \(G/T\) belongs in \(\varepsilon_{k+1}(\infty)\). Since \(G/T\) is nilpotent so soluble then by ([20], Theorem 3) there exist an integer \(c=c(k)\) depending only on \(k\) such that \((G/T)/Z_{c}(G/T)\) is finite. By a result in ([15], Theorem 1) \(\gamma _{c+1}(G/T)=\gamma _{c+1}(G)T/T\) is finite and so is torsion, and since \(T\) is torsion group , we deduce that \(\gamma _{c+1}(G)\) is torsion group. Therefore \(G\) is in the class of \(\tau N_{c}\)-group.
This completes the proof.

Lemma 2.13 Let \(G\) a finitely generated \(FN\)-group. Then

1. if \(G\) is in the class \(((FN_{k})F,\infty )^{\ast }\), then there exist an integer \(c=c(k)\) depending only on \(k\) such that \(G\) is in the class of \(FN_{c}\)-group.
2. if \(G\) is in the class \(((FC)F, \infty )^{\ast }\), then, \(G/Z_{2}(G)\) is finite and \(G\) is in the class of \(FN_{2}\)-groups.
3. if \(G\) is in the class \(((FN_{2})F, \infty )^{\ast }\), then, \(G\) is in the class of \(FN_{3}^{(2)}\)-groups and there exist an integer d such that \(G\) is \(FN_{3^{d-1}}\).

Proof. (1) Assume that \(G\) is finitely generated \(FN\)-group in the class \(((FN_{k})F, \infty)^{\ast}\). As \(G\) is \(FN\)-group, there exist a normal and finite subgroup \(F\) of \(G\) such that \(G/F\) is nilpotent quotient group. Since \(G/F\) finitely generated nilpotent group it has a torsion subgroup \(T/F\) of finite order. So the subgroup \(T\) of torsion elements of \(G\) is normal and finite in \(G\) and as the same way in the above theorem, we deduce by ([18], Lemma 6.33) that \(G/T\) is nilpotent torsion -free in the class \((N_{k},\infty )^{\ast }\subset \varepsilon_{k+1}(\infty)\) and according to ([20], Theorem 3) we found that there exist an integer \(c=c(k)\) depending only on \(k\) such that \((G/T)/Z_{c}(G/T)\) is finite. Also by ([15], Theorem 1) we find that \(\gamma _{c+1}(G)T/T\) is finite and as \(T\) is finite then \(\gamma _{c+1}(G)\) is finite too. Therefore \(G\) is in the class of \(FN_{c}\)-group.
(2) If \(G\) is in the class \(((FC)F,\infty )^{\ast }=((FA)F,\infty )^{\ast}=((FN_{1})F, \infty )^{\ast }\), then as in (i) \(G/T\) is in the class \((N_{1},\infty )^{\ast }\subset \varepsilon _{2}(\infty),\) by a result of Abdollahi [11] \((G/T)/Z_{2}(G/T)\) is finite and by ([15], Theorem 1) we find that \(\gamma _{3}(G)T/T\) is finite and as \(T\) is finite then \(\gamma _{3}(G)\) is finite too and as \(G\) is finitely generated then by [15],\(G/Z_{2}\) is finite. Therefore \(G\) is in the class of \(FN_{2}\)-group.
(3) For \(k=2\), as the same way in (1) we found that \(G/T\) is in the class \((N_{2},\infty )^{\ast }\) which is included in the class \(\varepsilon{3}(\infty )\), where \(\varepsilon_{3}(\infty )\) is the class of groups whose every infinite subset \(X\) contain two distinct elements \(x, y\) such that \([x,_{3}y]=1\). We deduce by (20, Theorem 1) that \(G/T\) is torsion-free in the class \(FN_{3}^{(2)}\) so G/T is in \(N_{3}^{(2)}\) and as the torsion subgroup \(T\) is finite, then \(G\) is \(FN_{3}^{(2)}\)-group. As \(G/T \) is torsion-free soluble group in \(N_{3}^{(2)}\subset \varepsilon _{3}\) (the 3-Engel group) then by Gruenberg [[18], Theorem 7.36 (1)], there exist integer d such that \(G/T\) is in the class \(N_{3^{d-1}}\) which gives that \(G\) is \(FN_{3^{d-1}}\). This completes the proof.

Theorem 2.14. Let \(G\) a finitely generated \(NF\)-group. Then

1. if \(G\) is in the class \(((FN_{k})F,\) \(\infty )^{\ast }\), then there exist an integer \(c=c(k)\) depending only on such that \(G\) is in the class of \(N_{c}F\)-groups.
2. if \(G\) is in the class of \(((FC)F, \infty )^{\ast }\)-groups, then, \(G\) is in the class of \(N_{2}F\)-group.
3. if \(G\) is in the class \(((FN_{2})F, \infty )^{\ast }\), then, \(G\) is in the class of \(N_{3}^{(2)}F\)-groups and there exist an integer \(d\) such that \(G\) is \(FN_{3^{d-1}}\).

Proof. (1) As the group \(G\) is \(NF\)- group, and then it contains a normal nilpotent subgroup \(N\) such that \(G/N\) is finite. As the subgroup \(N\) is finitely generated and nilpotent of finite index then \(N\) is polycyclic so by ([19], Theorem 5.4.15) there exist a normal subgroup \(M\) in \(N\) and poly-infinite cyclic hence torsion-free and of finite index in \(N\). Let \(K=M_{G}\) the core of the subgroup \(M\), so \(K\) is nilpotent torsion-free of finite index in \(G.\) Since the class \(((FN_{k})F, \infty )^{\ast }\) is closed under taking subgroups, then \(K\) is in this class too, so by (1) of lemma 2.13, we obtains that there exist an integer \(c=c(k)\) depending only on \(k\) such that \(K\) is \(FN_{c}\)-group and as \(K\) is torsion-free, it is \(N_{c}\)-group and so \(G\) is \(N_{c}F\)-group.
(2) Particularly for \(k=1\), we have \(((FC)F, \infty )^{\ast }=((FN_{1})F,\infty )^{\ast }\), in this case the subgroup \(K\) is a finitely generated torsion-free nilpotent group in the class \(((FN_{1})F, \infty )^{\ast }\) and according to (2) of lemma 2.13, we deduce that \(K\) is in the class \(FN_{2}\)-groups and as \(K\) is torsion-free, it is \(N_{2}\)-group of finite index in \(G\), this gives that \(G\) is \(N_{2}F\)-group.
(3) In particular for \(k=2\), we have the subgroup \(K\) in (1) is a finitely generated torsion-free nilpotent group in the class \(((FN_{2})F,\infty )^{\ast }\) and according to (3) of lemma 2.13, we deduce that \(K\) is in the class \(FN_{3}^{(2)}\)-groups and as \(K\) is torsion-free it is the class \(N_{3}^{(2)}\)-group and as \(G/K\) if finite this gives that \(G\) is in the class of \(N_{3}^{(2)}F\)-groups. As K is nilpotent torsion- free in the class \(N_{3}^{(2)}\) then it is in the class \(\varepsilon _{3}\) of 3-Enjel group, then by Gruenberg [18, Theorem 7.36 (1)], there exist integer \(d\) such that \(K\) is in the class \(N_{3^{d-1}}\) and so \(G\) is in the class \(N_{3^{d-1}}F\)-group.

Acknowledgments

The authors would like to thank the anonymous referees for their valuable comments and suggestions in advance.

Competing Interests

The author(s) do not have any competing interests in the manuscript.