УДК 512.74 Вестник СПбГУ. Сер. 1. 2013. Вып. 1 ON SUBNORMAL SUBGROUPS IN GENERAL SKEW LINEAR GROUPS∗ Bui Xuan Hai1 , Nguyen Van Thin2 1. University of Science, VNU-HCMC, Vietnam, Ph.D., bxhai@hcmus.edu.vn 2. University of Science, Vietnam, Ph.D., nguyenvanthin20@gmail.com This paper is dedicated to Professor Nikolai Vavilov on the occasion of his sixtieth birthday. 1. Introduction. Let D be a division ring with center F and the multiplicative group D∗ . In general, the structure of skew linear groups of degree n ≥ 1 over D is much less known. The most important results concerning such groups can be found in [17] and [18]. In this paper, we study properties of subnormal subgroups of GLn (D) for n ≥ 1. Let N be such a subgroup. We ask, under which conditions N must be in the center of GLn (D)? Let us recall some history. In 1905 Wedderburn proved the famous result (now known as Wedderburn’s “Little” Theorem): “Every ﬁnite division ring is commutative”. This means that ﬁnite groups can not occur as multiplicative groups of non-commutative division rings. This result stimulates many further investigations to generalize it. For instance, the well-known theorem of Kaplansky [13, (15.15), p. 259] states that if D is radical over F , then D = F ; L. K. Hua proved that if D∗ is solvable, then D is a ﬁeld,... There is a series of results of such a kind, known as commutativity theorems, even not only for division rings, but for associative rings in general. We refer to [14] for more details. Now, assume that N is a subnormal subgroup of D∗ = GL1 (D). C. J. Stuth [20] proved that if N is solvable, then N ⊆ F . Evidently, Stuth’s result is a broad generalization of Hua’s theorem. Recently, there are several results of such a kind appeared in the literature, see, for example [1, 7–10, 16]. Hence, it is natural to consider the same problem for subnormal subgroups of GLn (D) for n ≥ 2 and there are diﬀerent results concerning this problem obtained by several authors for last time (see, for example, [1–4, 15, 16], ...). In the present paper, we give in addition, other conditions for N to be central. Throughout this paper, D is a division ring with center F . We denote by D∗ the multiplicative group of D and by D := [D∗ , D∗ ] the derived subgroup of D∗ . We say that D is centrally ﬁnite if D is a ﬁnite dimensional vector space over F . If S is a non-empty subset of D, then we denote by F [S] and F (S) the subring and the division subring of D generated by S ∪ F respectively. If for any ﬁnite subset S of D, F (S) is a ﬁnite dimensional vector space over F , then we say that D is locally ﬁnite. An element a ∈ D is algebraic over F if there exists a non-zero polynomial f (t) ∈ F [t] such that f (a) = 0. A subset S of D is algebraic over F if every a ∈ S is algebraic 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 . ∗ This work is supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant 101.01-2011.16 and by Vietnam National University HoChiMinh City (VNU-HCMC) under grant B2012-18-31. c Bui Xuan Hai, Nguyen Van Thin, 2013 61 A non-empty subset S of D is radical over F if every element of S is radical over F . If G is any group, then Z(G) is the center of G. The similar symbol is used for a ring. In fact, if R is a ring, then Z(R) denotes the center of R. Recall that for n ≥ 2 we have [GLn (D), GLn (D)] = SLn (D) unless the case n = 2 and D = 2 (the ﬁeld of two elements). Note that the center of Mn (D) is the set F I = {xI : x ∈ F }, where I is the identity matrix and in the present paper we identify F I with F . 2. Results Lemma 2.1. Let D be an inﬁnite centrally ﬁnite division ring with center F . If N is a ﬁnitely generated normal subgroup of GLn (D), n ≥ 2, then N is contained in F . Proof. If N is not contained in F , then by [15, Theorem 11], SLn (D) ⊆ N . Since D is inﬁnite, SLn (D) is inﬁnite and so N is inﬁnite too. Now, by [2, Theorem 5], N is not ﬁnitely generated, a contradiction. Note that in [10], the notion of weakly locally ﬁnite division ring was ﬁrstly introduced (see also Deﬁnition 1.1 in [4]). In fact, 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 centrally ﬁnite. It was proved in [10] that every locally ﬁnite division ring is weakly locally ﬁnite and the converse is not true. Moreover, in [10], there is the example of a weakly locally ﬁnite division ring that is not even algebraic over its center. Of course, every locally ﬁnite division ring is algebraic (for instance, we note the longstanding conjecture [12], proposed by Kurosh in 1941, now known as Kurosh’s Problem for division rings which states that every algebraic division ring is locally ﬁnite). Thus, the class of weakly locally ﬁnite division rings strictly contains the class of locally ﬁnite division rings. In the following we mention some results about these rings. Theorem 2.2. Let D be an inﬁnite weakly locally ﬁnite division ring with center F . If N is a ﬁnitely generated subnormal subgroup of GLn (D), n ≥ 2, then N is contained in F . Proof. If N is non-central, then by [15, Theorem 11], SLn (D) ⊆ N . Suppose that N is generated by matrices A1 , . . . , Ak in GLn (D) and S is the set of all entries of all Aj , 1 ≤ j ≤ k. Since D is weakly locally ﬁnite, the division subring K of D generated by S is centrally ﬁnite. By Lemma 2.1, N ⊆ Z(GLn (K)), so N is abelian and consequently, SLn (D) is abelian too; a contradiction. Remark 1. Theorem 2.2 is not longer true for ﬁnite division rings. In fact, if D is ﬁnite then D is a ﬁeld and for n ≥ 2 we have [GLn (D), GLn (D)] = SLn (D) (unless the case n = 2 and D = 2 ) which is not contained in the center F = D. For the exceptional case, we note that G = GL(2, 2) ∼ = S3 and SL(2, 2) ∼ = A3 , so the conclusion is also obvious. In [7, Theorem 2.4], it was proved that if D is a division ring, algebraic over its center F , then every locally solvable subnormal subgroup of D∗ is central. In the following we show that this assertion remains also true for weakly locally ﬁnite division rings. Theorem 2.3. Let D be a weakly locally ﬁnite division ring and N be a subnormal subgroup of D∗ . If N is locally solvable, then N is contained in the center F of D. Proof. Take any two elements x and y from N and put D1 = F (x, y), F1 = Z(D1 ) and N1 = N ∩D1 . Since N1 is a subnormal locally solvable subgroup of D1∗ , by [7, Theorem 2.4], 62 N1 is contained in F1 . In particular, x and y commute with each other, so N is abelian. Now, by [19, 14.4.4, p. 440], N is contained in F . Recall Stuth’s theorem [20] which states that if N is a solvable subnormal subgroup of D∗ , then N is central. So, in view of [7, Theorem 2.4] and Theorem 2.3 above, we have a reason to pose the following conjecture. Conjecture 1. Let D be an arbitrary division ring. If N is a locally solvable subnormal subgroup of D∗ , then N is central. As we see in the following theorem, the same assertion holds in GLn (D), n ≥ 2 with only two exceptional “small” cases. Theorem 2.4. Let D be a division ring and N a locally solvable subnormal subgroup of GLn (D), n ≥ 2. Then, N is contained in the center F , unless the cases n = 2 and D = 2 or D = 3 . Proof. Suppose that N is not contained in F . Then, by [15, Theorem 11], SLn (D) ⊆ N , so SLn (D) is locally solvable. Case 1. n ≥ 3: For 1 ≤ i = j ≤ n and x ∈ D, denote by Ei,j (x) the matrix with x on the position (i, j), ones on the main diagonal and 0 elsewhere. Consider the set S = {E12 (1), E21 (1), E31 (1), E13 (1), E23 (1), E32 (1)} and suppose that H = S is the subgroup of SLn (D) generated by S. Then, H is solvable. On the other hand, since [Eij (1), Ejk (1)] = Eik (1) for any distinct i, j, k, we have S ⊆ H , hence H = H . The last equality shows that H is unsolvable, a contradiction. Case 2. n = 2: Subcase 1. CharD = p : Since PSL(2, p) is a non-abelian simple group, it is unsolvable. Hence, SL(2, p) is unsolvable ﬁnite subgroup of SLn (D) that is a contradiction. Subcase 2. CharD = 0 : Firstly, we note that the subgroup H = E12 (1/2), E21 (1/2), E12 (3), E21 (1/3) is solvable. By direct calculation, we have: 2 0 a= = E21 (−1/2)E12 (1)E21 (1)E12 (−1/2) ∈ H, 0 1/2 3 0 b= = E21 (−2/3)E12 (1)E21 (2)E12 (−1/3) ∈ H, 0 1/3 E12 (1) = [a−1 , E12 (−1)]3 [E12 (−1), b−1 ] ∈ H , E21 (1) = [b, E21 (1)][E21 (1), a]3 ∈ H , E12 (1/2) = [a−1 , E12 (−1/2)]E12 (−1) ∈ H , E12 (1/3) = [E12 (−1/3), b−1 ]E12 (3) ∈ H , E21 (1/2) = E21 (−1)[E21 (1/2), a] ∈ H , E21 (1/3) = E21 (3)[b, E21 (1/3)] ∈ H . Thus, H = H and this shows that H is unsolvable, a contradiction. 63 Remark 2. The cases n = 2 and D = 2 or D = 3 are really exceptional. In fact, the subgroups SL(2, 2) and SL(2, 3) are non-central solvable subnormal subgroups in GL(2, 2) and GL(2, 3) respectively. Corollary 2.5. Let D be a division ring and N a locally nilpotent subnormal subgroup of GLn (D), n ≥ 1. Then, N is contained in the center F , unless the cases n = 2 and D = 2 or D = 3 . Proof. In fact, the case n = 1 is Theorem 2.2 in [7]. For n ≥ 2, the conclusion follows from Theorem 2.4. Theorem 2.6. let D be a weakly locally ﬁnite division ring and suppose that N is a subnormal Engel subgroup of D∗ . Then, N is contained in the center F of D. Proof. Consider an arbitrary ﬁnite subset S of N and the subgroup H = S of N generated by S. If any two elements of S commute with each other, then H is abelian. Otherwise, put D1 = F (S) and denote by F1 the center of D1 . Then, it is clear that 2 ≤ m = [D1 : F1 ] < ∞. So, H can be viewed as a subgroup of GLm (F1 ). Since N is an Engel subgroup, H is an Engel subgroup too. By the result of Gruenberg [5], H coincides with its Hirsch-Plotkin radical, so H is nilpotent. Consequently, N is locally nilpotent. Now, by Corollary 2.5, N ⊆ F . Theorem 2.7. Assume that a division ring D with center F is diﬀerent from and n ≥ 2. If N is a subnormal Engel subgroup of GLn (D), then N ⊆ F. 2 and 3, Proof. Suppose that N ⊆ F . Then, by [15, Theorem 11], SLn (D) ⊆ N . Put ⎞ ⎞ ⎛ ⎛ 1 −1 2−1 ⎠ and y = ⎝ ⎠. 2 x=⎝ 1 In−2 In−2 For any integer k ≥ 1 we have ⎛ [x, k y] = ⎝ ⎞ 1 −3k 1 In−2 ⎠ = In . This shows that SLn (D) is not Engel subgroup, a contradiction. Hence, N ⊆ F . Theorem 2.8. Let D be a division ring with center F and suppose that N is a subnormal subgroup of GLn (D), n ≥ 2. Assume that D is not a ﬁeld of characteristic p, algebraic over p . If N is radical over F , then N ⊆ F . Proof. Suppose that N ⊆ F . Then, by [15, Theorem 11], SLn (D) ⊆ N . Case 1. D is non-commutative For any 1 = d ∈ D we have A = diag[d, 1, . . . , 1] ∈ SLn (D), so there exists some integer k such that Ak = diag[dk , 1, . . . , 1] = In . Hence, d is torsion and by [[14], Theorem 8], D ⊆ F . So, D∗ is solvable, and by [19, 14.4.4, p. 440], D is commutative, a contradiction. Case 2. D is commutative Suppose that CharD = 0. Then, we have A = diag[2, 1/2, 1, . . . , 1] ∈ SLn (D) ⊆ N , so A is radical over F . But this is a contradiction, since Ak ∈ F for any integer k ≥ 1. Hence, it remains to consider the case CharD = p > 0. Suppose that D is not algebraic over p . 64 Then, there exists some element a ∈ D such that ak = 1 for all k ≥ 1. Hence, the matrix diag[a, a−1 , 1, . . . , 1] is not radical over F although it is in SLn (D), a contradiction. Thus, in any case we have N ⊆ F and the proof is now complete. We note that in the theorem above the condition that D is not a ﬁeld of characteristic p, algebraic over p is really necessary (see Remark 3 below). Corollary 2.9. If n ≥ 2 and SLn (D) contains some proper subgroup of ﬁnite index, then D must be a ﬁeld of characteristic p, algebraic over p . Proof. Assume that N is a proper subgroup of ﬁnite index in SLn (D). Then, H = −1 N g is the normal subgroup of ﬁnite index in SLn (D), say, [SLn (D) : H] = g∈SLn (D) g r < ∞. By [6, 2.2.10, p. 79] and [6, 2.2. 13, p. 80], H ⊆ F . Since g r ∈ H, ∀g ∈ SLn (D), it follows that SLn (D) is radical over F . Now, the conclusion follows from Theorem 2.8. Corollary 2.10. If n ≥ 2 and SLn (D) contains some non-central F C-element, then D must be a ﬁeld of characteristic p, algebraic over p . Proof. Assume that x ∈ SLn (D) is a non-central F C-element. Then, [SLn (D) : CSLn (D) (x)] = r < ∞ and r ≥ 2. Now, the conclusion follows from Corollary 2.9. Corollary 2.11. Let D be a division ring with center F and suppose that N is a subnormal subgroup of GLn (D), n ≥ 1. Assume that D is not a ﬁeld of characteristic p, algebraic over p . If N is locally ﬁnite, then N ⊆ F . Moreover, if n = 1, then the conclusion holds without any constrain for D. Proof. If n ≥ 2, then the conclusion follows from Theorem 2.8. The case n = 1 follows from [8, Theorem 2.3]. Theorem 2.12. Let D be a division ring with center F and suppose that N is a subnormal subgroup of GLn (D), n ≥ 1. Assume that D is not a ﬁeld of characteristic p, algebraic over p . If N is an F C-group, then N ⊆ F . Moreover, if n = 1, then the conclusion holds without any constrain for D. Proof. If n = 1, then N ⊆ F by [8, Theorem 3.1]. Now, for n ≥ 2, suppose that N ⊆ F . Then, by [15, Theorem 11], SLn (D) ⊆ N , so SLn (D) is an F C-subgroup. Case 1. D is non-commutative d Take an element d = 1 in D and consider a matrix A = in SLn (D). In−1 Since SLn (D) is an F C-subgroup, the following subset is ﬁnite a −1 In−1 d In−1 a In−1 −1 a da |a ∈ D = In−1 a ∈ D . Hence, D is an F C-subgroup and by [8, Theorem 3.1], D ⊆ F , so D∗ is solvable, and consequently D is commutative, a contradiction. 65 Case 2. D is commutative Assume that CharD = 0. Then, the following set ⎛ 1 ⎝ k ⎞−1 ⎛ 1 ⎠ In−2 ⎛ ⎝ ⎞⎛ 2 1/2 2 = ⎝ −3k/2 1/2 In−2 In−2 1 ⎠⎝ k ⎞ ⎞ ⎠= 1 In−2 ⎠ k ∈ is inﬁnite that is a contradiction since SLn (D) is an F C-subgroup. Hence, it remains to consider the case CharD = p. By supposition, D is not algebraic over p , so there exist a, b ∈ D∗ such that a is not algebraic over p and b = b−1 . Then, the following set ⎞−1 ⎛ ⎞⎛ ⎞ ⎛ b 1 1 ⎠ ⎝ ⎠ ⎝ ai 1 ⎠ i ∈ ⎝ ai 1 b−1 In−2 In−2 ⎞ In−2 ⎛ b ⎠ i ∈ is inﬁnite, that is a contradiction. = ⎝ ai (b−1 − b) b−1 In−2 Thus, in any case we have a contradiction; hence N ⊆ F , so the proof is now complete. Remark 3. 1) The condition that D is not a ﬁeld of characteristic p, algebraic over p in Theorem 2.8 and Theorem 2.12 is really necessary. In fact, suppose that F is a ﬁeld of characteristic p, algebraic over p and n ≥ 2. Consider an arbitrary matrix A = (aij ) ∈ SLn (F ) and assume that K is the subﬁeld of F generated by all aij over p . Then K is a ﬁnite ﬁeld and it follows that SLn (K) is a ﬁnite group. Hence, A is a torsion element and consequently, SLn (F ) is a torsion non-central normal subgroup of GLn (F ). 2) If F is a ﬁnite ﬁeld, then the conclusion in Corollary 2.11 is no longer true. In fact, this can be seen from the fact that for n ≥ 2, SLn (F ) is a ﬁnite non-central normal subgroup of GLn (F ). Corollary 2.13. Let D be a division ring with center F and suppose that N is a subnormal subgroup of D∗ . If a ∈ N is an F C-element in N , then a ∈ F . Proof. Clearly we can suppose that D is non-commutative. The set S = {ag |g ∈ N } is ﬁnite, so we can write S = {ag1 , . . . , agt } for some positive integer t. Clearly, the subgroup H = S is normal in N . For x ∈ H, we have the following expression x = aε1 gj1 . . . aεk gjk , with εi = ±1 and gji ∈ S. Choose k as small as possible in the above expression of x. Then, the number of elements xh , h ∈ H does not exceed tk , so H is an F C-group. Now, by [8, Theorem 3.1], H ⊆ F and so a ∈ F . 66 References 1. Akbari S., Mahdavi-Hezavehi M., Mahmudi M. G. Maximal subgroups of GL1 (D) // J. Algebra. 1999. Vol. 217. P. 422–433. 2. Akbari S., Mahdavi-Hezavehi M. Normal subgroups of GLn (D) are not ﬁnitely generated // Proceedings of the AMS. 1999. Vol. 128, no. 6. P. 1627–1632. 3. Akbari S., Ebrahimian R., Momenaee Kermani H., Salehi Golseﬁdy A. 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