On almost subnormal subgroups in division rings
aa r X i v : . [ m a t h . R A ] J a n ON ALMOST SUBNORMAL SUBGROUPS IN DIVISION RINGS
HUYNH VIET KHANH AND BUI XUAN HAI
Abstract.
Let D be a division ring with center F , and G an almost subnormalsubgroup of D ∗ . In this paper, we show that if G is locally solvable, then G ⊆ F . Also, assume that M is a maximal subgroup of G . It is shown thatif M is non-abelian locally solvable, then [ D : F ] = p for some prime number p . Moreover, if M is locally nilpotent then M is abelian. Introduction
There has been recently a great deal of interest in algebraic structure of themultiplicative group D ∗ of a division ring D . For a self-contained account of thedevelopment on this area of study, we refer to [15]. It is not much to say that thestarting point is the Wedderburn Little Theorem stating that any finite divisionring is a field. This famous theorem is the subject for many generalizations. Manyauthors paid attention on the question that how far D ∗ , and more generally itssubnormal subgroups, from being abelian. On this line, the well-known result ofHua states that if D ∗ is solvable then D is a field. Some special types of subnormalsubgroups such as nilpotent, solvable, and locally nilpotent have been examined.For instance, Stuth [26, Theorem 6 (iii)] proved that every solvable subnormal sub-group of D ∗ is central, i.e, it is contained in the center F of D . In [18], Huzurbazarshowed that this result remains true if the word ‘solvable’ is replaced by ‘locallynilpotent’. In [11, Theorem 2.3], the authors extended the result of Hua by prov-ing that if D ∗ is locally solvable, then D is a field. It is clear that if G is locallynilpotent or solvable, then it is locally solvable. However, the converse does nothold: there exists a division ring which contains a locally solvable subgroup that isneither solvable nor locally nilpotent (see [25, 1.4.13]). In view of [26, Theorem 6(iii)] and [11, Theorem 2.3], it is natural to ask whether every locally solvable sub-normal subgroup of D ∗ is central (see [12, Conjecture 1]). It was shown in [13]that the question has the positive answer in the case when D is algebraic over F .Recently, the positive answer to this question in general case have been obtained in[2]. One of our purposes in this paper is to extend this result for almost subnormalsubgroups. We shall prove that in a division ring D with center F every locallysolvable almost subnormal subgroup is central.Let us recall the notion of almost subnormal subgroups. Following Harley [14],a subgroup H of a group G is called almost subnormal in G if there is a finitesequence of subgroups H = H ≤ H ≤ · · · ≤ H n = G, Key words and phrases. division ring; locally solvable subgroup; almost subnormal subgroup.2010
Mathematics Subject Classification. in which either [ H i +1 : H i ] is finite or H i is normal in H i +1 for 0 ≤ i ≤ n − H is a subnormal subgroup of G , then it is almost subnormal. But wedo not have the converse. In [3] and [22], there are examples of almost subnormalsubgroups in division rings that are not subnormal. Recently, almost subnormalsubgroups in skew linear groups have been studied in [3], [8], [10], [22].Note that in the case of arbitrary center F , T. H. Dung [4] has recently provedthat any locally solvable almost subnormal subgroup G of D ∗ is central, provided G is algebraic over F .In another direction, maximal subgroups of D ∗ , and more generally of subnormalsubgroups, were considered. The authors whose study in this direction often preferto examine how big these maximal subgroups are, that is, how such subgroupsreflect the algebraic structure of D . Some kinds of such subgroups such that abelian,nilpotent, and solvable subgroups were considered. It was shown in [6] that anynilpotent maximal subgroup of D ∗ is abelian. The result was then generalized bythe authors in [23] that any nilpotent maximal subgroup of a subnormal subgroupis also abelian. Unfortunately, the result is no longer true if the word ‘nilpotent’is replaced by ‘solvable’. In fact, in the multiplicative group H ∗ of the divisionring of real quaternions H , the set C ∗ ∪ C ∗ j forms a non-abelian solvable maximalsubgroup (see [1]). In general, it was showed that if the multiplicative group D ∗ of a division ring D contains a non-abelian solvable maximal subgroup, then D iscyclic algebra of prime degree over the center F (see [5]). In the beginning of 2019,the works in [7] and [20] show that this result can be extended to the subnormalsubgroups of D ∗ . For precisely, assume that G is a subnormal subgroup of D ∗ andthat M is a maximal subgroup of G . It was shown that if M is non-abelian solvable,then [ D : F ] = p for some prime p . It is natural to ask that can we obtain theanalogous results in the case ‘locally nilpotent’ or ‘locally solvable’ ? Relating tothis question, the authors in [9] showed that if M is non-abelian locally solvable,then D is a cyclic of prime degree, provided M ′ is algebraic over F . It was shownin [19] that this results still hold if the i -th derived subgroup M ( i ) of M is algebraicover F for some i , instead of M ′ . In this paper, we confirm that the answer is ‘yes’in general setting. Let G be an almost subnormal subgroup of D ∗ , and assumethat M is a maximal subgroup of G . We show that if M is a non-abelian locallysolvable maximal subgroup, then D is cyclic of prime degree over F . Also, if M islocally nilpotent maximal subgroup, then M is abelian.Through out this paper, for a ring R with the identity 1 = 0, the symbol R ∗ stands for the group of units of R . If D is a division ring with the center F and S ⊆ D , then F [ S ] and F ( S ) denotes respectively the subring and the divisionsubring of D generated by F ∪ S . For a group G and a positive integer i , the symbol G ( i ) is the i -th derived subgroup of G . If G is a group and H, K are subgroups of G , then H K ( G ) is the normalizer of H in K , that is H K ( G ) = K ∩ N G ( H ). If A isa ring or a group, then Z ( A ) denotes the center of A .2. Locally solvable almost subnormal subgroups are central
We need the following lemma from [33].
Lemma 2.1 ([33, 1.4]) . Let D = E ( G ) be a division ring generated by its locallynilpotent subgroup G and its division subring E such that E ≤ C D ( G ) . Set H = N D ∗ ( G ) . Denote the maximal -subgroup of G by Q . Then one of the followingholds: LMOST SUBNORMAL SUBGROUPS 3 (i) If G = Q · C G ( Q ) where Q is quaternion of order 8, then H/GE ∗ ∼ = Sym (3) × Y for some abelian group Y .(ii) If Q is non-abelian with < | Q | < ∞ , then H/GE ∗ has an abelian subgroup Y of index in H/GE ∗ at most .(iii) In all other cases H/GE ∗ is abelian. In the above lemma if we take E = F , where F is the center of D , then we havethe following result. Lemma 2.2.
Let D be a division ring with center F , and G a locally nilpotentsubgroup of D ∗ such that F ( G ) = D . If H = N D ∗ ( G ) , then H/GF ∗ is solvable. (cid:3) The well-known Cartan-Brauer-Hua Theorem says that any division subring K of a division ring D normalized by D ∗ is either contained in the center F of D orequals to D . The theorem was later generalized by many authors. One of them wasC. J. Stuth [26], who proved that the theorem also true if D ∗ is replaced by a non-central subnormal subgroup [26, Theorem 1]. Recently, the authors in [3] extendedthe result to the case almost subnormal subgroups, provide that F is infinite. Thefollowing theorem says that we no need to restrict any condition on F . Theorem 2.3.
Let D be a division ring with center F , and K a division subring of D . Assume that G is a non-central almost subnormal subgroup of D ∗ . If g − Kg ⊆ K for any g ∈ G , then either K ⊆ F or K = D .Proof. If F is infinite, then the result is exactly [3, Theorem 3.10]. In the case of F is finite, we will show that G contains a non-central subnormal subgroup of D ∗ .By hypothesis, there exists a finite chain of subgroups G = G ≤ G ≤ · · · ≤ G n = D ∗ , for which either [ G i +1 : G i ] is finite or G i E G i +1 . We will prove first that G containsa non-central subnormal subgroup by induction on n . By induction hypothesis, G possesses a non-central subnormal subgroup N . There are two possible cases: Case 1: G = G E G . In this case N ∩ G is a non-central normal subgroupof N (see [26, Lemma 4]). It follows that N ∩ G is a subnormal subgroup of D ∗ contained in G . Case 2: [ G : G ] < ∞ . If C = Core G ( G ), then C is a normal subgroup of finiteindex in G contained in G . If C ⊆ F , then fact that F is finite implies that C is afinite group. This yields that G is finite, and so is N . It follows by [16, Theorem8] that N ⊆ F , a contradiction. We may therefore assume that C F . By thesame arguments used in Case 1, we conclude that C ∩ N is a non-central subnormalsubgroup of D ∗ .Now, let H be a non-central subnormal subgroup of D ∗ contained in G . Since K is normalized by G , it is also normalized by H , and the result follows by [26,Theorem 1]. (cid:3) Lemma 2.4.
Let D be a division ring with center F . If G is a solvable almostsubnormal subgroup of D ∗ , then G ⊆ F . HUYNH VIET KHANH AND BUI XUAN HAI
Proof.
Trivially G satisfies a non-trivial group identity. If F is infinite, then theresult follows from [22, Theorem 2.2]. In the case when F is finite, if G is notcontained in F , then the proof of Theorem 2.3 implies that G contains a solvablenon-central subnormal subgroup, which is contained in F . But this contradicts to[26, Theorem 6(iii)]. (cid:3) Lemma 2.5.
Let G be a group. If G/Z ( G ) is locally finite, then so is G ′ .Proof. Let A = h a , a , . . . , a n i be a finitely generated subgroup of G ′ . Then wemay write a i = [ x i , y i ] · [ x i , y i ] · · · [ x i k , y i k ] , where x i j , y i j ∈ G for 1 ≤ i ≤ n . If N is the subgroup of G generated by all x i j , y i j ,then A ⊆ N ′ . By hypothesis, we conclude that N Z ( G ) /Z ( G ) is finite. It followsby Schur’s Theorem that N ′ = ( N Z ( G )) ′ is finite ([24, 10.1.3, p.278]); note that Z ( G ) ⊆ Z ( N Z ( G )). This implies that A is finite. (cid:3) Lemma 2.6. If D is a divsision ring such that D ( k ) is locally nilpotent for some k ≥ , then D is commutative.Proof. Since D ( k ) is normal in D ∗ , by [28, Theorem 1], D ( k ) /Z ( D ( k ) ) is locallyfinite, so D ( k +1) is locally finite by Lemma 2.5. By [16, Theorem 8], we concludethat D ( k +1) ⊆ F , which implies that D is solvable. By a result of Hua [17], itfollows that D is commutative. (cid:3) Proposition 2.7.
Let D be a division ring with center F . If G is a locally nilpotentalmost subnormal subgroup of D ∗ , then G ⊆ F .Proof. Assume by contradiction that G is not contained in F . If F is finite, then G contains a locally nilpotent non-central subnormal subgroup of D ∗ , which contra-dicts to [18]. Thus, G must be contained in F in this case. It remains the case F is infinite. Since G is an almost subnormal subgroup of D ∗ , there is a finite chainof subgroups G = G ≤ G ≤ · · · ≤ G n = D ∗ , for which either [ G i +1 : G i ] is finite or G i E G i +1 , for 0 ≤ i ≤ n −
1. If [ G : G ] isfinite, then C := Core G ( G ) = T x ∈ G x − G x is a normal subgroup of finite indexin G contained in G . It follows that s := [ G : C ] is finite. If C ⊆ F , then x s ! y s ! x − s ! y − s ! = 1 is a group identity of G . It follows by [22, Theorem 2.2] that G = G ⊆ F , a contradiction. Therefore C is not contained in F . By replacing G by C if necessary, we can assume that G is normal in G .Since G F , it follows by Theorem 2.3 that F ( G ) = D . If we set H = N D ∗ ( G ), then Lemma 2.2 says that H/G F ∗ is solvable. Since G is assumedto be normal in G , we conclude that G ⊆ H . It follows that G /G ∩ G F ∗ ∼ = G F ∗ /G F ∗ is also solvable. Thus, there exists an integer k for which G ( k )1 ⊆ G F ∗ , which is a locally nilpotent group. If G ( k )1 ⊆ F , then G is solvable. Ac-cording to Lemma 2.4, we conclude that G ⊆ F , a contradiction. Therefore, wemay assume that G ( k )1 F .Now G ( k )1 is a non-central locally nilpotent normal subgroup of G . By re-placing G and G by G and G ( k )1 in the preceding paragraph, we conclude that G ( k )2 is locally nilpotent, for some k . By finite steps, we obtain that D ( k n ) is LMOST SUBNORMAL SUBGROUPS 5 locally nilpotent for some k n . This fact together with Lemma 2.6 implies that D iscommutative, a contradiction. (cid:3) Lemma 2.8.
Let D be a division ring with center F , and G a subgroup of themultiplicative group D ∗ of D . If G is locally finite, then F ( G ) = F [ G ] .Proof. Asumme that 0 = x ∈ F [ G ]. Then, x = f g + f g + · · · + f n g n for some f i ∈ F and g i ∈ G . The subgroup H = h g , g , · · · , g n i is finite, so F [ H ] is adomain which is a finite dimensional vector space over F . This implies that F [ H ]is a division ring, so F ( H ) = F [ H ]. Hence, x − ∈ F [ H ] ⊆ F [ G ]. Consequently, F ( G ) = F [ G ]. (cid:3) Lemma 2.9.
Let D be a division ring with center F , and G a locally finite subgroupof D ∗ for which F ( G ) = D . Then, D is a locally finite division ring.Proof. By Lemma 2.8, D = F [ G ], so for any finite subset { x , x , . . . , x k } ⊆ D , wecan write x i = f i g i + f i g i + · · · + f i s g i s , where f i j ∈ F and g i j ∈ G . Let H = (cid:10) g i j : 1 ≤ i ≤ k, ≤ j ≤ s (cid:11) be the subgroupof G generated by all g i j . Since G/G ∩ F ∗ ∼ = GF ∗ /F ∗ is locally finite, the group HF ∗ /F ∗ is finite. Let { y , y , . . . , y t } be a transversal of F ∗ in HF ∗ and set S = F y + F y + · · · + F y t . Then, clearly S is a division ring that is finite dimensional over F containing { x , x , · · · , x k } . It follows that F ( x , x , · · · , x k ) is a finite dimensional vectorspace over F . (cid:3) Lemma 2.10.
Let D be a division ring with center F . If G is a locally finite almostsubnormal subgroup of D ∗ , then G ⊆ F .Proof. Assume by contradiction that G F . It follows by Theorem 2.3 that F ( G ) = D . In view of Lemma 2.9, we conclude that D is a locally finite divisionring. According to [22, Theorem 4.3], we conclude that G contains a non-cyclic freesubgroup. But this is impossible since G is locally finite. (cid:3) Lemma 2.11.
Every locally solvable periodic group is locally finite.Proof . Let G be a locally solvable periodic group, and H a finitely generatedsubgroup of G . Then, H is solvable with derived series of length n ≥
1, say,1 = H ( n ) E H ( n − E · · · E H ′ E H. We shall prove that H is finite by induction on n . For if n = 1, then H is a finitelygenerated periodic abelian group, so it is finite. Suppose n >
1. It is clear that
H/H ′ is a finitely generated periodic abelian group, so it is finite. Hence, H ′ isfinitely generated. By induction hypothesis, H ′ is finite, and as a consequence, H is finite. (cid:3) Remark 1.
For a group G , let τ ( G ) be the unique maximal periodic normalsubgroup of G , and B ( G ) a subgroup of G such that B ( G ) /τ ( G ) is the Hirsch-Plotkin radical of G/τ ( G ). It is clear that both τ ( G ) and B ( G ) are characteristicsubgroups of G . The group B ( G ) contains the Hirsch-Plotkin radical of G . So, if B ( G ) is locally nilpotent, then it is coincided with the Hirsch-Plotkin radical of G .If G is locally solvable, then by Lemma 2.11, τ ( G ) is locally finite. Moreover, if G HUYNH VIET KHANH AND BUI XUAN HAI is locally nilpotent, then τ ( G ) is the set of all elements of finite order of G (see [24,12.1.1]). Lemma 2.12 ([29, Point 20]) . Let R = F [ G ] be an algebra over the field F thatis a domain, where G is a locally solvable subgroup of the group of units of R suchthat B ( G ) = F ∗ ∩ G . Then, R is an Ore domain. Moreover, if D is the skew fieldof fractions of R , then N D ∗ ( G ) = GF ∗ . Lemma 2.13.
Let D be a division ring with infinite center F . If G is a locallysolvable almost subnormal subgroup of D ∗ , then B ( G ) ⊆ F .Proof . It follows by Lemma 2.10 that τ ( G ) is contained in F . Take any finitelygenerated subgroup H of B ( G ). Since B ( G ) /τ ( G ) is locally nilpotent, it followsthat Hτ ( G ) /τ ( G ) is nilpotent. Thus, we have [[ H, H ] , . . . , H ] ⊆ τ ( G ) ⊆ F , fromwhich it follows that H is nilpotent. Therefore, B ( G ) is a locally nilpotent group.Since B ( G ) is normal in G , it is an almost subnormal subgroup of D ∗ . If followsby Proposition 2.7 that B ( G ) ⊆ F . (cid:3) Lemma 2.14.
Let D be a division ring with center F . If G is a locally solvablenon-central almost subnormal subgroup of D ∗ , then F ( G ) = D and N D ∗ ( G ) = GF ∗ .Proof . By Theorem 2.3 we conclude that F ( G ) = D . If we set R = F [ G ],then R is an Ore domain by [30, Corollary 24]. Thus, the skew field of fractionsof R is coincided with D . It follows by Lemma 2.13 that B ( G ) ⊆ F ∩ G , hence B ( G ) /τ ( G ) ⊆ ( F ∩ G ) /τ ( G ). Since ( F ∩ G ) /τ ( G ) is an abelian normal subgroupof G/τ ( G ), the maximality of B ( G ) /τ ( G ) in G/τ ( G ) implies that B ( G ) /τ ( G ) =( F ∩ G ) /τ ( G ), from which we have B ( G ) = F ∩ G . Finally, by Lemma 2.12, wehave N D ∗ ( G ) = GF ∗ . (cid:3) Theorem 2.15.
Let D be a division ring with center F . If G is a locally solvablealmost subnormal subgroup of D ∗ , then G ⊆ F .Proof . Assume by contradiction that G F . If F is finite, then the proof ofTheorem 2.3 says that G contains a locally solvable non-central subnormal subgroupof D ∗ , which contracts to [2]. Now, assume that F is infinite. Since G is an almostsubnormal subgroup of D ∗ , there is a finite chain of subgroups G = G ≤ G ≤ · · · ≤ G n = D ∗ , for which either [ G i +1 : G i ] is finite or G i E G i +1 , for 0 ≤ i ≤ n −
1. By the sameargument used in the first paragraph of the proof of Proposition 2.7, we may assumethat G is normal in G . It follows by Lemma 2.14 that N D ∗ ( G ) = G F ∗ , whichis a locally solvable group. Since G ⊆ N D ∗ ( G ), we conclude that G is a locallysolvable group. After finite steps, we obtain that D ∗ is locally solvable. Accordingto [11, Theorem 2.1], we conclude that D is commutative, a contradiction. (cid:3) Locally solvable maximal subgroups
Let R be a ring, S a subring of R , and G a subgroup of the group of units of R normalizing S such that R = S [ G ]. Suppose that N = G ∩ S is a normal subgroupof G and R = L t ∈ T tS , where T is some (and hence any) transversal T of N to G .Then, we say that R is a crossed product of S by G/N (see [31] or [25, p.23]). Forthe convenience of readers, we gather a number of theorems of B. A. F Wehrfritz,which will be used in the proofs of our results.
LMOST SUBNORMAL SUBGROUPS 7
Lemma 3.1 ([27, 2.5]) . Let R = F [ G ] be an F -algebra, where F is a field and G is a locally nilpotent group of units of R , such that for every finite subset X of R there is a finitely generated subgroup Y of G with F [ Y ] prime and containing X .Let τ ( G ) be the unique maximal periodic normal subgroup of G , and Z/τ ( G ) thecenter of G/τ ( G ) . Then, R is a crossed product of F [ Z ] by G/Z . Lemma 3.2 ([28, 7]) . Let R = F [ G ] be an F -algebra, where F is a field and G is a locally solvable subgroup of the group of units of R such that for every infinitesubgroup X of G the left annihilator of X − in R is { } . Let B ( G ) be a subgroupof G such that B ( G ) /τ ( G ) is the Hirsch-Plotkin radical of G/τ ( G ) . Then R is acrossed product of F [ B ( G )] by G/B ( G ) . Lemma 3.3 ([31, 3.2]) . Let R be a ring, J a subring of R , and H ≤ K subgroupsof the group of units of R normalizing J such that R is the ring of right quotients of J [ H ] ≤ R and J [ K ] is a crossed product of J [ B ] by K/B for some normal subgroup B of K . Then K = HB . For a group G , we denote by Z ( G ), which is defined by Z ( G ) /Z ( G ) = Z ( G/Z ( G )),the second center of G . Lemma 3.4 ([31, Theorem 1.1(c)]) . Let G be a h P, L i A -subgroup of GL n ( D ) suchthat the subalgebra F [ N ] of M n ( D ) is a prime ring for every characteristic subgroup N of G . Denote by τ ( G ) the unique maximal locally finite normal subgroup of G .If τ ( G ) ⊆ Z ( G ) , then F [ G ] is a crossed product of F [ A ] by G/A , for some abeliancharacteristic subgroup of G . Remark 2.
In the above lemma, the notation h P, L i A stands for a class of groups,in which A denotes the class of abelian groups and P and L the poly and localoperators ([31]). It is clear that the class of groups h P, L i A contains that of locallysolvable groups. Lemma 3.5 ([32, Proposition 4.1]) . Let D = E ( A ) be a division ring generated byits metabelian subgroup A and its division subring E such that E ≤ C D ( A ) . Set H = N D ∗ ( A ) , B = C A ( A ′ ) , K = E ( Z ( B )) , H = N K ∗ ( A ) = H ∩ K ∗ , and let T = τ ( B ) be the unique maximal periodic normal subgroup of B .(i) If T has a quaternion subgroup Q = h i, j i of order with A = QC A ( Q ) , then H = Q + AH , where Q + = h Q, j, − (1 + i + j + ij ) / i . Also, Q is normalin Q + and Q + / h− , i ∼ = Aut Q ∼ = Sym (4) .(ii) If T is abelian and contains an element x of order not in the center of B ,then H = h x + 1 i AH .(iii) In all other cases, H = AH . Lemma 3.6.
Let D be a division ring with center F , and G an almost subnormalsubgroup of D ∗ . Assume that M is a non-abelian locally solvable maximal subgroupof G . If A E M , then either A is abelian or F ( A ) = D .Proof. Since A E M , we have M ⊆ N G ( F ( A ) ∗ ) ⊆ G . The maximality of M in G implies that either N G ( F ( A ) ∗ ) = M or N G ( F ( A ) ∗ ) = G . If the first case occurs,then A E F ( A ) ∗ ∩ G is almost subnormal in F ( A ) ∗ contained in M . Since M islocally solvable, so is A . It follows by Theorem 2.15 that A is contained in thecenter of F ( A ), so A is abelian. If N G ( F ( A ) ∗ ) = G , then F ( A ) is a division subringof D normalized by G . It follows by Theorem 2.3 that either A ⊆ F (and hence A is abelian) or F ( A ) = D . (cid:3) HUYNH VIET KHANH AND BUI XUAN HAI
The proof of the following proposition is a simple modification of the proof of [9,Theorem 3.3], so it should be omitted.
Proposition 3.7.
Let D be a division ring with center F , and G an almost sub-normal subgroup of D ∗ . If M is a non-abelian metabelian maximal subgroup of G ,then [ D : F ] < ∞ . Lemma 3.8.
Let H ≤ G be groups. If H is a characteristic subgroup of G , thenso is C G ( H ) .Proof. For any ϕ ∈ Aut( G ) and x ∈ C G ( H ), our task is to show that ϕ ( x ) ∈ C G ( H ).Take an arbitrary element h ∈ H . Since ϕ ( H ) = H , there exists h ′ ∈ H such that ϕ ( h ′ ) = h . Now, we have hϕ ( x ) = ϕ ( h ′ ) ϕ ( x ) = ϕ ( h ′ x ) = ϕ ( xh ′ ) = ϕ ( x ) ϕ ( h ′ ) = ϕ ( x ) h, which implies that ϕ ( x ) ∈ C G ( H ). (cid:3) Theorem 3.9.
Let D be a division ring with center F , and G an almost subnormalsubgroup of D ∗ . If M is a non-abelian solvable maximal subgroup of G , then thefollowing hold:(i) There exists a maximal subfield K of D such that K/F is a finite Galoisextension with
Gal(
K/F ) ∼ = M/K ∗ ∩ G ∼ = Z p and [ D : F ] = p , for someprime number p .(ii) The subgroup K ∗ ∩ G is the F C -center. Also, K ∗ ∩ G is the Hirsch-Plotkinradical of M . For any x ∈ M \ K , we have x p ∈ F and D = F [ M ] = L pi =1 Kx i .Proof. First, we prove that [ D : F ] < ∞ . Since M is non-abelian, Lemma 3.6 saysthat F ( M ) = D . Also, we may suppose that M is solvable with derived length s ≥
2. Therefore, there exists such a series1 = M ( s ) E M ( s − E M ( s − E · · · E M ′ E M. If we set A = M ( s − , then A is a non-abelian metabelian normal subgroup of M .In view of Lemma 3.6, we conclude that F ( A ) = D . It follows that Z ( A ) = F ∗ ∩ A and F = C D ( A ). Set H = N D ∗ ( A ), B = C A ( A ′ ), K = F ( Z ( B )), H = H ∩ K ∗ , and T = τ ( B ) to be the maximal periodic (and hence locally finite) normal subgroup of B . Since A ′ is a characteristic subgroup of A , by Lemma 3.8, B is a characteristicsubgroup of A . By Remark 1, T is a characteristic subgroup of B , hence T is alsoa characteristic subgroup of A . It is clear that H is an abelian group, so in viewof Lemma 3.5, we have three possible cases: Case 1: T is not abelian.Again by Lemma 3.6, we have F ( T ) = D . The local finiteness of T together withLemma 2.9 implies that D is a locally finite division ring. Since M is solvable, itcontains no non-cyclic free subgroups. By [10, Theorem 3.1], it follows [ D : F ] < ∞ . Case 2: T is abelian and contains an element x of order 4 not in the center of B = C A ( A ′ ).It is clear that x is not contained in F . Because x is of finite order, the field F ( x ) is algebraic over F . It was proved (see the proof of [32, Theorem 1.1, p. 132])that h x i is a 2-primary component of T , hence it is a characteristic subgroup of T . LMOST SUBNORMAL SUBGROUPS 9
Consequently, h x i is a normal subgroup of M . Thus, all elements of the set x M := { m − xm | m ∈ M } ⊆ F ( x ) have the same minimal polynomial over F . This implies | x M | < ∞ , so x is an F C -element, and consequently, [ M : C M ( x )] < ∞ . Setting C = Core M ( C M ( x )), then C E M and [ M : C ] is finite. By Lemma 3.6, either C isabelian or F ( C ) = D . The latter case implies that x ∈ F , a contradiction. Thus, C is abelian. If we set K = F ( C ), then the finiteness of M/C implies that K is asubfield of D , over which D is finite dimensional. This fact yields that [ D : F ] < ∞ . Case 3: H = AH .Since A ′ ⊆ H ∩ A , we have H/H ∼ = A/A ∩ H is abelian, and hence H ′ ⊆ H .Since H is abelian, H ′ is abelian too. Moreover, M ⊆ H , it follows that M ′ is alsoabelian. In other words, M is a metabelian group, and the conclusions follow fromProposition 3.7.By what we have proved, we conclude that n := [ D : F ] < ∞ . Since M issolvable, it contains no non-cyclic free subgroups. In view of [10, Theorem 3.1], wehave F [ M ] = D , there exists a maximal subfield K of D containing F such that K/F is a Galois extension, N G ( K ∗ ) = M , K ∗ ∩ G is the Fitting normal subgroupof M and it is the F C -center, and
M/K ∗ ∩ G ∼ = Gal( K/F ) is a finite simple groupof order [ K : F ]. Since M/K ∗ ∩ G is solvable and simple, one has M/K ∗ ∩ G ∼ =Gal( K/F ) ∼ = Z p for some prime number p . Therefore, [ K : F ] = p and [ D : F ] = p .For any x ∈ M \ K , if x p F , then by the fact that F [ M ] = D , we conclude that C M ( x p ) = M . Moreover, since x p ∈ K ∗ ∩ G , it follows that h x, K ∗ ∩ G i ≤ C M ( x p ).In other words, C M ( x p ) is a subgroup of M strictly containing K ∗ ∩ G . Because M/K ∗ ∩ G is simple, we have C M ( x p ) = M , a contradiction. Therefore x p ∈ F .Furthermore, since x p ∈ K and [ D : K ] r = p , we conclude D = L p − i =1 Kx i . Itremains to prove that K ∗ ∩ G is the Hirsch-Plotkin radical of M . Note that weare in the case that K ∗ ∩ G ⊆ M , we conclude that K ∗ ∩ G = K ∗ ∩ M . Thus, wehave M/K ∗ ∩ M ∼ = M K ∗ /K ∗ ∼ = Z p . Let H be the Hirsch-Plotkin radical of M .Then H K ∗ /K ∗ ≤ M K ∗ /K ∗ , thus either H ⊆ K ∗ or H K ∗ = M K ∗ . The first caseimplies that H ⊆ K ∗ ∩ G , so H = K ∗ ∩ G ; we are done. If the second case occurs,then F [ H K ∗ ] = F [ M ] = D . It follows that H K ∗ is a locally nilpotent absolutelyirreducible subgroup of D ∗ , hence it is center-by-locally finite by [28, Theorem 1].It is clear that the center of H K ∗ is contained in F ∗ . This yields that H K ∗ /F ∗ is locally finite, from which it follows that K/F is a non-trivial radical Galoisextension. According to [21, 15.13], we conclude that D is algebraic over a finitesubfield. But then Jacobson’s Theorem ([21, 13.11]) says that D is commutative,a contradiction. (cid:3) Theorem 3.10.
Let D be a division ring with center F , and G an almost subnormalsubgroup of D ∗ . If M is a locally nilpotent maximal subgroup of G , then M isabelian.Proof. Assume by contradiction that M is non-abelian. We claim that [ D : F ] < ∞ . Let T = τ ( M ) be the unique maximal periodic (hence locally finite) normalsubgroup of M , and Z/T be the center of
M/T . Since T is normal in M , it followsby Lemma 3.6 that either F ( T ) = D or T is abelian. If the first case occurs, thenLemma 2.9 says that D is a locally finite division ring. According to [10, Theorem3.1], we have [ D : F ] < ∞ ; we are done. Now assume that T is abelian. Thisfact implies that Z is a solvable group with the derived series length of s ≥ say. Since T and Z/T is a characteristic subgroup of M and M/T respectively, weconclude that Z is a characteristic subgroup of M . Again in view of Lemma 3.6,either F ( Z ) = D or Z is abelian. Let us consider the following two possible cases: Case 1: F ( Z ) = D .Since D is non-commutative, it follows that Z is non-abelian. Thus, we maysuppose that it is solvable with derived length s ≥
2. Therefore, there exists sucha series 1 = Z ( s ) E Z ( s − E Z ( s − E · · · E Z ′ E Z E M. If we set A = Z ( s − , then A is a non-abelian metabelian normal subgroup of M .In view of Lemma 3.6, we have F ( A ) = D . It follows that Z ( A ) = F ∗ ∩ A and F = C D ( A ). Set H = N D ∗ ( A ), B = C A ( A ′ ), K = F ( Z ( B )), H = H ∩ K ∗ , and T = τ ( B ) to be the unique maximal periodic normal subgroup of B . Then H isan abelian group, and T is an abelian (contained in T ) characteristic subgroup of B , and hence of A (Remark 1). It follows that T is normal in M . By replacing T by T in the proof of Theorem 3.9, we also have [ D : F ] < ∞ . Case 2: Z is abelian.Let N be the maximal subgroup of M with respect to the property: N is anabelian normal subgroup of M containing Z . We shall show that M/N is a simplegroup. For, let P be a normal subgroup of M properly containing N . Clearly P is non-abelian. By Lemma 3.6, we have F ( P ) = D . Since P is locally nilpotent,it follows by [30, Corollary 24] that F [ P ] is an Ore domain whose skew field offractions is coincided with D . In view of Lemma 3.1, we conclude that F [ M ] is acrossed product of F [ Z ] by M/Z . It follows by Lemma 3.3 that M = P Z = P ;recall that Z ⊆ P . This fact shows that M/Z is simple. Note that M = Z since M is non-abelian. Because M/Z is locally nilpotent, we conclude that
M/Z is finiteof prime order ([24, 12.5.2, p.367]). If we set L = F ( Z ), then L is a subfield of D ,over which D is finite dimensional. This fact yields [ D : F ] < ∞ .By what we have proved, we conclude that n := [ D : F ] < ∞ . We know that D ⊗ F D op ∼ = M n ( D ). Therefore, we may view M as a subgroup of GL n ( F ) toconclude that it is a solvable group. According to Theorem 3.9, we conclude that M is distinguish from its Hirsch-Plotkin radical, which contrasts to the fact that M is locally nilpotent. (cid:3) Theorem 3.11.
Let D be a division ring with center F , and G an almost subnormalsubgroup of D ∗ . If M is a non-abelian locally solvable maximal subgroup of G , thenthe following hold:(i) There exists a maximal subfield K of D such that K/F is a finite Galoisextension with
Gal(
K/F ) ∼ = M/K ∗ ∩ G ∼ = Z p and [ D : F ] = p , for someprime number p .(ii) The subgroup K ∗ ∩ G is the F C -center. Also, K ∗ ∩ G is the Hirsch-Plotkinradical of M . For any x ∈ M \ K , we have x p ∈ F and D = F [ M ] = L pi =1 Kx i .Proof. First, we show that [ D : F ] < ∞ . Since M is non-abelian, Lemma 3.6 saysthat F ( M ) = D . Let T = τ ( M ) be the unique maximal periodic normal subgroupof M . If T is non-abelian, then by the same arguments used in the beginning of LMOST SUBNORMAL SUBGROUPS 11 the proof of Theorem 3.10 we conclude that [ D : F ] < ∞ ; we are done. We maytherefore suppose that T is abelian. There are two possible cases. Case 1: T F .From the field theory, F ( T ) is an algebraic field extension of F . Take x ∈ T \ F ,and set x M := { m − xm | m ∈ M } . Since F ( T ) is normalized by M , we have x M ⊆ F ( T ). Thus, the all elements of the set x M have the same minimal polynomialover F ⊆ F ( T ), hence x M is finite. In other words, x is an F C -element of M . Bythe same argument used in Case 2 of the proof of Theorem 3.9, we conclude that[ D : F ] < ∞ . Case 2: T ⊆ F .Let B = B ( M ) be the subgroup of M such that B/T is the Hirsch-Plotkinradical of the group
G/T . As we see in the proof of Lemma 2.13, the group B islocally nilpotent. It follows that B is the the Hirsch-Plotkin radical of M (Remark1). If M = B , then M is locally nilpotent and thus it is abelian by Theorem 3.10,a contradiction. We may therefore assume that M = B . We claim that M/B isa simple group. In fact, let C be a normal subgroup of M properly containing C . It is clear that C is non-abelian. It follows by Lemma 3.6 that F ( C ) = D .According to Lemma 3.2, we conclude that F [ M ] is a crossed product of F [ B ] by M/B . Moreover, by [30, Corollary 24], it follows that F [ C ] is an Ore domain whoseskew field of fractions is coincided with D . Therefore, we may apply Lemma 3.3to conclude that M = BC , which implies that M = C . Thus, the group M/B issimple, as claimed. Again by [24, 12.5.2, p.367], we have
M/B is a finite groupof prime order. In view of Lemma 3.6, either B is abelian or F ( B ) = D . Let usconsider the following two subcases. Subcase 2.1: B is abelian.If we set L = F ( B ), then L is a subfield of D . Since F ( M ) = D and | M/B | < ∞ ,we have [ D : L ] r < ∞ . This implies that [ D : F ] < ∞ , and we are done. Subcase 2.2: F ( B ) = D .Since we are in the case T = τ ( M ) ⊆ F ∗ ∩ M ⊆ Z ( M ), we may apply Lemma3.4 and Remark 2 to conclude that F [ M ] is a crossed product of F [ A ] by M/A , forsome abelian characteristic subgroup A of M . Since AT /T is an abelian normalsubgroup of
M/T , we have
AT /T ⊆ B/T ; recall that
B/T is the Hirsch-Plotkinradical of
M/T . This implies that A ⊆ B . By Lemma 3.3, we conclude that M = BA = B . This contrasts to the fact that B is a proper subgroup of M .By what we have proved, we conclude that n := [ D : F ] < ∞ . We know that D ⊗ F D op ∼ = M n ( F ). Thus, we may view M as a subgroup of GL n ( F ), to concludethat M is solvable. The results follow by Theorem 3.9. (cid:3) References [1] S. Akbari, R. Ebrahimian, H. Momenaei Kermani, A. Salehi Golsefidy, Maximal subgroups ofGL n ( D ), J. Algebra 259 (2003) 201-225.[2] L. Q. Danh, H. V. Khanh, Locally solvable subnormal and quasinormal subgroups of divisionrings, arXiv:1903.11216 [math.RA][3] T. T. Deo, M. H. Bien, B. X. Hai, On division subrings normalized by almost subnormalsubgroups in division rings, Period. Math. Hung. (2019); DOI: 10.1007/s10998-019-00282-5.[4] T. H. Dung, A note on locally soluble almost subnormal subgroups in divsion rings, Int. J.Group Theory (2019), DOI: 10.22108/IJGT.2019.116399.1546.[5] H. R. Dorbidi, R. Fallah-Moghaddam, M. Mahdavi-Hezavehi, Soluble maximal subgroups inGL n ( D ), J. Algebra Appl. (6)10 (2011) 1371-1382.[6] R. Ebrahimian, Nilpotent maximal subgroups of GLn(D), J. Algebra 280 (2004) 244248.[7] R. Fallah-Moghaddam, Maximal subgroups of SL n ( D ), J. Algebra 531 (2019) 70-82.[8] B. X. Hai, H. V. Khanh, and M. H. Bien, Generalized power central group identities in almostsubnormal subgroups of Algebra i Analiz 34:1 (2019) 225-239 (Russian), English transl. in St.Petersburg Math. J. (2019) (accepted).[9] B. X. Hai and N. A. Tu. On multiplicative subgroups in division rings, J. Algebra Appl. 15(3)(2016), 1650050 (16 pages).[10] B. X. Hai and H. V. Khanh, Free subgroups in maximal subgroups of skew linear groups,Internat. J. Algebra Comput. 29(3) (2019) 603-614.[11] B. X. Hai, D. V. P. Ha, On locally soluble maximal subgroups of the multiplicative group ofa division ring, Vietnam J. Math. 38(2) (2010) 237- 247.[12] B. X. Hai and N. V. Thin, On subnormal subgroups in general skew linear groups, VestnikSt. Petersburg University. Mathematics, 46 (2013) 43-48.[13] B. X. Hai and N. V. Thin, On locally nilpotent subgroups of GL ( D ), Comm. Algebra 37(2009) 712-718.[14] B. Hartley, Free groups in normal subgroups of unit groups and arithmetic groups, Contemp.Math. 93 (1989) 173-177.[15] R. Hazrat, M. Mahdavi-Hezavehi and M. Motiee, Multiplicative groups of division rings,Math. Proc. R. Ir. Acad. 114A (2014) 37-114.[16] I. N. Herstein. Multiplicative commutators in division rings. Israel J. Math. 31 (1978) 180-188.[17] L. K. Hua, On the multiplicative group of a field, Acad. Sinica Science Record (1950).[18] M. S. Huzurbazar, The multiplicative group of a division ring, Soviet Math. Dokl. (1960)1433-1435.[19] H. V. Khanh, On locally solvable subgroups in division rings, arXiv:1903.11216v3 [math.RA].[20] H. V. Khanh, B. X. Hai, A note on solvable maximal subgroups in subnormal subgroups ofGL n ( D ), arXiv:1809.00356v2 [math.RA].[21] T. Y. Lam, A First Course in Noncommutative Rings, 2nd edn, GTM 131, Springer-Verlag,New York, 2001.[22] N. K. Ngoc, M. H. Bien, B. X. Hai, Free subgroups in almost subnormal subgroups of generalskew linear groups, Algebra i Analiz 28(5) (2016) 220-235, translation in St. Petersburg Math.J. 28(5) (2017) 707-717.[23] M. Ramezan-Nassab and D. Kiani, Nilpotent and polycyclic-by-finite maximal subgroups ofskew linear groups, J. Algebra 399 (2014) 269-276.[24] D. J. S. Robinson, A Course in the Theory of Groups, 2nd edn, Springer, 1995.[25] M. Shirvani and B. A. F. Wehrfritz, Skew Linear Groups, Cambridge Univ. Press, 1986.[26] C. J. Stuth, A generalization of the Cartan-Brauer-Hua Theorem, Proc. Amer. Math. Soc.15(2) (1964) 211-217.[27] B. A. F. Wehrfritz, Locally nilpotent skew linear groups, Proc. Edinburgh Math. Soc. 29(1986) 101-113.[28] B. A. F. Wehrfritz, Locally nilpotent skew linear group II, Proc. Edinburgh, Math. So. 30(3)(1987) 423-426.[29] B. A. F. Wehrfritz, Soluble and locally soluble skew linear groups, Arch. Math. 49 (1987)379-388.[30] B. A. F. Wehrfritz, Goldie subrings of Artinian rings generated by groups, Q. J. Math. Oxford40 (1989) 501-512. LMOST SUBNORMAL SUBGROUPS 13 [31] B. A. F. Wehrfritz, Crossed product criteria and skew linear groups, J. Algebra 141 (1991)321353.[32] B. A. Wehrfritz, Normalizers of nilpotent subgroups of division rings, Q. J. Math. 58 (2007)127-135.[33] B. A. F. Wehrfritz, Normalizers of subgroups of division rings, J. Group Theory 11 (2008)399413.
E-mail address : [email protected]; [email protected]@gmail.com; [email protected]