The motivation of this study comes from a question posed by Herstein [13] in 1978: Is it true that every subnormal subgroup of the multiplicative group of a division ring D which is radical over the center F of D is central? Herstein himself showed that the conditions "subnormal" and "normal" are equivalent [14]. At the present, the question is affirmatively answered for the following particular cases:

• • for periodic subgroups [13, Theorem 8];

• • for a division ring with uncountable center [14, Theorem 2];

• • for a division ring finite dimensional over the center [9, Theorem 1];

• • for normal subgroups in a division ring of type 2, where the center F is replaced by an arbitrary proper division subring K of D [8, Theorem 3.2].

Motivated by the results obtained in [8][Theorem 3.2], the authors posed a more general question: For a division ring D, given a normal subgroup N of D^*, is N contained in the center F of D, provided it is radical over some proper division subring K of D? In this paper, we replace the assumption "normal" with "quasinormal". In fact, for a division ring D with center F, we show that every quasinormal subgroup of the multiplicative group of D which is radical over some proper division subring is central in case of one of the following conditions:

• (1) D is weakly locally finite;

• (2) F is uncountable;

• (3) D is the Mal'cev-Neumann division ring.

Recall that, in the theory of division rings, there are several classical constructions of new division rings from given ones. One of such structures is the class of Mal'cev-Neumann division rings, were completely presented in [18] by Neumann who used Mal'cev's ideas in [17]. Mal'cev-Neumann division rings have a vast number of applications. For example, they were used to construct examples of non-crossed product division algebras [2, 10, 11], to describe the multiplicative group of group rings of ordered groups, [15, Corollary 14.24], etc. The problems describe the properties of Mal'cev Neumann division rings and their special cases have been studied in several papers. For instance, it was proved that there are free group algebras in the Mal'cev-Neumann division rings [19]; there are free symmetric group algebras in division rings generated by poly-orderable groups [6]. Also, Amitsur and Tignol determined abelian Galois subfields of the Mal'cev-Neumann division rings with G finite and D a field in [22]. In [4], the authors introduced weakly locally finite division rings. Recall that a division ring D is called weakly locally finite if for every finite subset S of D, the division subring of D generated by S is finite dimensional over its center. Also in [4], by using the general structure of the Mal'cev-Neumann division rings, they gave an example to prove that the class of weakly locally finite division rings is strictly contained in the class of division rings that is finite dimensional over its center.

The paper is organized as follows: In Section 2, we prove that N is central in the case where D has the uncountable center and that N is a normal subgroup of D^* which is radical over a proper division subring of D. This is an important result for replacing the "normal subgroup" by the "quasinormal subgroup" in the next sections. In Section 3, by considering the similar result of the Herstein conjecture and by considering D to be either a weakly locally finite division ring or a division ring with uncountable center, we prove that every quasinormal subgroup of D^* which radical over a proper division subring of D is central. In Section 4, we also give some results regarding quasinormal subgroups of D^* which is radical over a proper division subring of D, as an interesting example for the class of these subgroups. In Section 5, we investigate quasinormal subgroups of the multiplicative group D* of the Mal'cev-Neumann division ring D, we also show that every quasinormal subgroup of D* which is radical over a proper division subring of D is central.

The symbols and notations we use in this paper are standard that can be found in the literature on division rings and in the cited items of the presented paper.

Recall that a division ring D with center F is said to be a division ring of type 2 if for every two elements x,y∈D, the division subring F(x,y) generated by x,y over F is a finite dimensional vector space over F. An element x ∈ D is radical over F if there exists some positive integer n(x) depending on x such that xn(x)∈F. A subset S of D is radical over F if every element of S is radical over F. If we replace F by some division subring L of D, then we have the notion of radicality over L.

In [8, Theorem 3.2], it was shown that every normal subgroup of the multiplicative group of a division ring of type 2 which is radical over its proper division subring is central. The proof of [8, Theorem 3.2] is based on the properties of a division subring generated by two elements over F. We use this idea to prove the following theorem.

Theorem 2.1. Let D be a division ring with center F and N a normal subgroup of D^* which is radical over a proper division subring of D. Assume that S is a finite subset of D and K=F(S) is the division subring of D generated by S over F. Then, N∩K is radical over the center Z(K) of K.

Proof. Let L be a proper division subring of D such that N is radical over L. If |S|=1, then K is a field. Thus, let |S|≥2. We can assume that S={x1,…,xk}, where k≥2. Suppose a∈N∩K. We claim that, for any x in S, there exists a positive integer n so that anx=xan.

Case 1: x∈L

Clearly, we can assume a+x≠0 and x≠±1. Consider the elements α=(a+x)a(a+x)−1 and β=(x+1)a(x+1)−1. Since N is normal in D^* and radical over L, there exist some positive integers m₁ and m₂ such that

With n1=m1m2, we have

There are two possible subcases for a given element a∈N.

Subcase 1.1: a∈L

Then,

This implies

Hence,

Since x∈L and αn1−βn1,an1(a−1)+βn1−αn1a∈L, from the last equality we must have αn1−βn1=0. Now, from (3), we get an1=αn1=βn1. Recall that βn1=(x+1)an1(x+1)−1. This implies that an1x=xan1.

Subcase 1.2: a∈L

Since a is radical over L, there exists a positive integer m such that b:=am∈L. According to Subcase 1.1, there exists a positive integer n₁ such that bn1x=xbn1. Hence, amn1x=xamn1.

Case 2: x∈L

Take c∈D\L. According to Case 1, there exist positive integers m₁ and m₂ such that am1c=cam1 and am2(x+c)=(x+c)am2. Hence, am1m2x=xam1m2.

Thus, the claim is proved.

Therefore, for each i=1,2,...,k, there exists a positive integer n_i such that anixi=xiani. Now, if n=n1n2…nk, then anxi=xian for all i=1,2,...,n. This implies that an∈Z(K), that is, a is radical over Z(K).

Using Theorem 2.1, we can now extend some earlier results. In [14], Herstein proved that in a division ring D with uncountable center F, if a normal subgroup of D^* is radical over F then it is central. In the following theorem, we will show that F can be replaced by an arbitrary proper division subring of D.

Theorem 2.2. Let D be a division ring with uncountable center and N a normal subgroup of D^*. If N is radical over a proper division subring of D, then it is central.

Proof. Let F be the center of D. Assume that N is non-central. Take x∈N and y∈D such that xy≠yx and put K=F(x,y). By Theorem 2.1, N∩K is radical over the center Z(K) of K. Since Z(K) contains F, Z(K) is uncountable. By [14, Theorem 2], N∩K⊆Z(K). In particular, xy=yx, a contradiction. Hence, N is central.

Based on the definition of the weakly locally finite division rings in [4], in this section, we will show a similar result to a conjecture of Herstein for quasinormal subgroups. First, we use the following result on the relationship between quasinormal subgroups, radical subgroups, and subnormal subgroups.

Recall that a subgroup H of a group G is called subnormal in G if there exists a series of r+1 subgroups

A quasinormal subgroup (or permutable subgroup) is a subgroup H of a group G that commutes (permutes) with every other subgroup K of G with respect to the product of subgroups, i.e. HK=KH.

Lemma 3.1. Let G be a group and N a quasinormal subgroup of G. Then, either N is subnormal in G or G is radical over N.

Proof. This lemma is [3, Lemma 6].

Next, combining [4, Theorem 11] and Theorem 2.2, we have the following result for the normal subgroup of the multiplicative group of a division ring.

Theorem 3.2. Let D be a division ring with center F, L a proper division subring of D, and N a normal subgroup of D^* which is radical over L. If either F is uncountable or D is weakly locally finite, then N is central.

Proof. The result is obtained from [4, Theorem 11] and Theorem 2.2.

For a group G, the normal core CoreG(H) of a subgroup H in G is the largest normal subgroup of G that is contained in H (or equivalently, the intersection of all conjugates of H), i.e.,

To prove the main result of this section, we need the following results from [7, Theorem 1], [21, Theorem 4], and [5, Theorem B].

Lemma 3.3. Let G be a group and H a subnormal subgroup of G. If H is a quasinormal subgroup of G, then H/CoreG(H) is a solvable group.

Proof. This lemma is a part of [7, Theorem 1].

Lemma 3.4.([21, Theorem 4]) Let D be a division ring. Then, all solvable subnormal subgroups of D^* are central

The following is an interesting result of Faith in [5] regarding a division ring radical over its proper division subring.

Lemma 3.5.([5, Theorem B]) Let D be a division ring and L be a proper division subring of D. If D is radical over L, then D is commutative.

Now we show the main result of this section.

Theorem 3.6. Let D be either a weakly locally finite division ring or a division ring with uncountable center, and N a quasinormal subgroup of D^* which radical over a proper division subring of D. Then, N is central.

Proof. Let F be the center of D and assume that L is a proper division subring of D such that N is radical over L. Since N is quasinormal in D^*, by Lemma 3.1, either N is a subnormal subgroup in D^* or D^* is radical over N.

Case 1: N is subnormal in D^*. Since CoreD*(N)⊆N, CoreD*(N) is radical over L. Furthermore, CoreD*(N) is a normal subgroup of D^*, so CoreD*(N)⊆F, by Theorem 3.2. In particular, CoreD*(N) is solvable. On the other hand, due to Lemma 3.3, we get that N/CoreD*(N) is also solvable, so N is solvable. Then, by Lemma 3.4, we have N is central.

Case 2: D^* is radical over N. Since N is radical over L, we obtain that D is radical over L. Then, according to Lemma 3.5, D is commutative. Consequently, N is central.

In both cases, N is central. The proof is now complete.

In this section, we will give some results regarding quasinormal subgroups of D^* which are radical over a proper division subring of D, as an interesting example for the class of these subgroups. First, we have the following lemma.

Lemma 4.1. Let D be a division ring with center F, and assume that N is a normal subgroup of D^*. If N is radical over F, then for any a ∈ N with a² ∈ F, we have a ∈ F.

Proof. Since a∈ N, for any x∈D*, there exists a positive integer n such that (axa−1x−1)n∈F. If a2∈F, by [13, Theorem 6], a∈ F.

Let D be a division ring with center F. An element a in D is called algebraic over F if a is a root of a nonzero polynomial over F. Additionally, the minimal polynomial of a, denoted ma(t), is the monic polynomial of of lowest degree such that ma(a)=0. For convenience of use, we give the following two lemmas which are based on similar ideas from the work [16] of M. Mahdavi-Hezavehi and S. Akbari-Feyzaabaadi with some slight modifications.

Lemma 4.2. Let D be a division ring with center F, and assume that N is a normal subgroup of D^* which is radical over F. If the minimal polynomial of an element a∈N has degree n, then an∈F.

Proof. By replacing N by the subgroup generated by N∪F*, so without loss of generality, we assume that N contains F^*. Let a∈N. We assume that am∈F for some positive m and n is the degree of ma(t). We must show that an∈F. Indeed, observe that n=[F(a):F] so that by [16, Lemma 1], there exists c∈F(a)*∩D′ such that NF(a)/F(a)=can. Then, c=NF(a)/F(a)a−n which implies that cm=NF(a)/F(a)ma−mn=λ∈F. Since NF(a)/F(c)=1, we obtain that 1=NF(a)/F(c)mn=λn=cmn. On the other hand, since N contains F, we have c∈ N. By [13, Theorem 9], c∈F, it follows that an=c−1NF(a)/F(a)∈F.

Lemma 4.3. Let D be a division ring with center F, and assume that N is a normal subgroup of D^*. If N is radical over F, then for any a∈N with a3∈F, we have a ∈ F.

Proof. We can suppose that D is noncommutative. Assume that a∈N such that a3∈F. Let f(t) be the minimal polynomial of a over F. Then, the degree of f(t) is less than or equal to 3. If the degree of f(t) is 2, then by Lemma 4.2, a² ∈ F. Due to Lemma 4.1, a∈ F. Now we assume that the degree of f(t) is 3. By Lemma 4.2 again, the minimal polynomial of a is f(t)=t3−λ, with λ∈F. By Wedderburn's Theorem,

Thus, a+bab−1+dad−1=0. Let α=bab−1a−1 and β=dad−1a−1, leads to 1+α+β=0. By the fact that α and β are in N, it follows that α and β are radical over F. Let m be the degree of the minimal polynomial of α. Then, m=[F(α):F]. Because 1+α+β=0, m=[F(β):F]. By Lemma 4.2, αm∈F and βm∈F, so we have αm∈F and (1+α)m∈F. Moreover, since m is the degree of the minimal polynomial of α on F, we get m=1 or char(D)=p>0 and p∣m. If the latter happens, then by putting m=pq we get αm=(αq)p∈F, so by [14, Lemma 2] we obtain αq∈F, which contradicts the minimality of m. So m=1, that is, α∈F. We claim that α =1. Assume that α≠1. By the fact that bab−1=αa, one has a3=(bab−1)3=α3a3, so α3=1. Therefore, ab−3a−1=(ab−1a−1)3=α3b−3=b−3, so we have ab3=b3a. Put D1=F(a,b) the division subring of D generated by a,b over F. Observe that α≠1, so that ab≠ba. It implies that D₁ is noncommutative. Let F₁ be the center of D₁. Observe that b3a=ab3, so b3∈F1. Because bab−1=αa, and α,b3∈F1, every element in D₁ may be written as

Hence, D₁ is finite dimensional over the subfield F₁(a) of D₁ generated by F₁. It is well known that D₁ satifies a polynomial identity and, as a corollary, D is centrally finite. The claim is shown. Since N is radical over F, it follows that N∩D1 is radical over F₁. Thus, according to [4, Theorem 10], N∩D1⊆F1. In particular, a∈F1, i.e. ab=ba, a contradiction. The claim is shown, that is, α=1. Similarly, β=1. Hence, char(D)=3. Since a3∈F, so by applying [13, Lemma 1] we obtain a ∈ F

Lemma 4.4. Let D be a division ring with center F, and assume that N is a normal subgroup of D^* which is radical over a proper division subring of D. Then CD(a)=CD(a2)=CD(a3) for any a∈N.

Proof. Clearly, CD(a)⊆CD(a2). Now let b∈CD(a2), and put K=F(a,b). By Theorem 2.1, N∩K is radical over the center Z(K) of K. Observe that a2∈Z(K), so a∈ Z(K) by Lemma 4.1. In particular, ab=ba. Hence, CD(a)=CD(a2).

Using Lemma 4.3, by the similar way, we can get CD(a)=CD(a3).

Now we can prove an analogue of [16, Corollary 3] for a quasinormal subgroup which is radical over a proper division subring.

Theorem 4.5. Let D be a division ring with center F and N a quasinormal subgroup of D^*. Assume that L is a proper division subring of D and for every x∈ N, there exist positive integers m,n such that x2n3m∈L. Then, N is central.

Proof. With similar reasons to those in the proof of Theorem 3.6, we only need to consider the case of N is a normal subgroup of D^*. It suffices to prove that C_D(a)=D for any a∈N. For any a∈N,b∈D, by repeating the argument in the proof of Thereom 2.1, there exist positive integers m and n such that a2m3nb=ba2m3n. It implies that b∈CD(a2n3m). By Lemma 4.4, CD(a)=CD(a2n3m), which implies that b∈CD(a). Thus, CD(a)=D for any a∈N.

Theorem 4.6. Let D be a division ring, L a proper division subring of D, and N a quasinormal subgroup of D*. If there exists a positive integer d such that for every x∈N, xn∈L for some n≤d, then N is central.

Proof. With similar reasons to those in the proof of Theorem 3.6, we only need to consider the case of N is a normal subgroup of D^*. Let F be the center of D. For any x,y ∈ N, put K=F(x,y) and M=N∩K. Observe that for any a ∈ M, by the same argument as in the proof of Theorem 2.1, we can choose some n≤(d!)3 such that an∈Z(K). By [1, Theorem 1], M is abelian, so xy=yx. Hence, N is abelian, and by [12], N is central.

Recall that a division ring D with center F is said to be a division ring of type 2 if for every two elements x,y∈D, the division subring F(x,y) generated by x,y over F is a finite dimensional vector space over F. An element x ∈ D is radical over F if there exists some positive integer n(x) depending on x such that xn(x)∈F. A subset S of D is radical over F if every element of S is radical over F. If we replace F by some division subring L of D, then we have the notion of radicality over L.

In [8, Theorem 3.2], it was shown that every normal subgroup of the multiplicative group of a division ring of type 2 which is radical over its proper division subring is central. The proof of [8, Theorem 3.2] is based on the properties of a division subring generated by two elements over F. We use this idea to prove the following theorem.

Theorem 2.1. Let D be a division ring with center F and N a normal subgroup of D^* which is radical over a proper division subring of D. Assume that S is a finite subset of D and K=F(S) is the division subring of D generated by S over F. Then, N∩K is radical over the center Z(K) of K.

Proof. Let L be a proper division subring of D such that N is radical over L. If |S|=1, then K is a field. Thus, let |S|≥2. We can assume that S={x1,…,xk}, where k≥2. Suppose a∈N∩K. We claim that, for any x in S, there exists a positive integer n so that anx=xan.

Case 1: x∈L

Clearly, we can assume a+x≠0 and x≠±1. Consider the elements α=(a+x)a(a+x)−1 and β=(x+1)a(x+1)−1. Since N is normal in D^* and radical over L, there exist some positive integers m₁ and m₂ such that

With n1=m1m2, we have

There are two possible subcases for a given element a∈N.

Subcase 1.1: a∈L

Then,

This implies

Hence,

Since x∈L and αn1−βn1,an1(a−1)+βn1−αn1a∈L, from the last equality we must have αn1−βn1=0. Now, from (3), we get an1=αn1=βn1. Recall that βn1=(x+1)an1(x+1)−1. This implies that an1x=xan1.

Subcase 1.2: a∈L

Since a is radical over L, there exists a positive integer m such that b:=am∈L. According to Subcase 1.1, there exists a positive integer n₁ such that bn1x=xbn1. Hence, amn1x=xamn1.

Case 2: x∈L

Take c∈D\L. According to Case 1, there exist positive integers m₁ and m₂ such that am1c=cam1 and am2(x+c)=(x+c)am2. Hence, am1m2x=xam1m2.

Thus, the claim is proved.

Therefore, for each i=1,2,...,k, there exists a positive integer n_i such that anixi=xiani. Now, if n=n1n2…nk, then anxi=xian for all i=1,2,...,n. This implies that an∈Z(K), that is, a is radical over Z(K).

Using Theorem 2.1, we can now extend some earlier results. In [14], Herstein proved that in a division ring D with uncountable center F, if a normal subgroup of D^* is radical over F then it is central. In the following theorem, we will show that F can be replaced by an arbitrary proper division subring of D.

Theorem 2.2. Let D be a division ring with uncountable center and N a normal subgroup of D^*. If N is radical over a proper division subring of D, then it is central.

Proof. Let F be the center of D. Assume that N is non-central. Take x∈N and y∈D such that xy≠yx and put K=F(x,y). By Theorem 2.1, N∩K is radical over the center Z(K) of K. Since Z(K) contains F, Z(K) is uncountable. By [14, Theorem 2], N∩K⊆Z(K). In particular, xy=yx, a contradiction. Hence, N is central.

The authors would like to thank the referee for his/her useful suggestions. This work was written while the second author was working at the Vietnam Institute for Advanced Study in Mathematics (VIASM). The author would like to thank the institute for providing a fruitful research environment and working condition.

This research is funded by University of Architecture Ho Chi Minh City under grant number KHCB 03 - NCKH 22.