Journal of Algebra 259 (2003) 201–225 www.elsevier.com/locate/jalgebra Maximal subgroups of GLn(D) S. Akbari,a,b,∗ R. Ebrahimian,a H. Momenaee Kermani,a and A. Salehi Golsefidy a a Department of Mathematical Sciences, Sharif University of Technology, PO Box 11365-9415, Tehran, Iran b Institute for Studies in Theoretical Physics and Mathematics, Tehran, Iran Received 16 January 2002 Communicated by Jan Saxl Abstract In this paper we study the structure of locally solvable, solvable, locally nilpotent, and nilpotent maximal subgroups of skew linear groups. In [S. Akbari et al., J. Algebra 217 (1999) 422–433] it ∗ has been conjectured that if D is a division ring and M a nilpotent maximal subgroup of D ,thenD is commutative. In connection with this conjecture we show that if F [M]\F contains an algebraic ∗ ∗ element over F ,thenM is an abelian group. Also we show that C ∪ C j is a solvable maximal subgroup of real quaternions and so give a counterexample to Conjecture 3 of [S. Akbari et al., J. Algebra 217 (1999) 422–433], which states that if D is a division ring and M a solvable maximal ∗ subgroup of D ,thenD is commutative. Also we completely determine the structure of division rings with a non-abelian algebraic locally solvable maximal subgroup, which gives a full solution to both cases given in Theorem 8 of [S. Akbari et al., J. Algebra 217 (1999) 422–433]. Ultimately, we extend our results to the general skew linear groups. 2002 Elsevier Science (USA). All rights reserved. Keywords: Division rings; Skew linear groups; Nilpotent; Irreducible; Maximal subgroups Introduction The theory of linear groups over division rings of a fairly general type has been intensively developed in recent decades. The most important results concerning skew linear groups can be found in the interesting book of Shirvani and Wehrfritz (1986) [19]. * Corresponding author. E-mail addresses: [email protected] (S. Akbari), [email protected] (R. Ebrahimian), [email protected] (H. Momenaee Kermani), golsefi[email protected] (A. Salehi Golsefidy). 0021-8693/02/$ – see front matter 2002 Elsevier Science (USA). All rights reserved. PII:S0021-8693(02)00549-5 202 S. Akbari et al. / Journal of Algebra 259 (2003) 201–225 Since typically maximal subgroups are large, their algebraic properties carry over to those of the whole group. In this article we are interested in investigating what happens whenever a maximal subgroup of a general skew linear group admits some specific algebraic property. There are many articles devoted to investigations on skew linear groups include one-dimensional case, i.e., GL1(D), the multiplicative group of a division ring D. Recently the structure of maximal subgroups of division rings have been studied; for instance, see [1–3]. In [3] the following two conjectures have appeared. Conjecture A. Let D be a division ring and M a solvable maximal subgroup of D∗.Then D is commutative. Conjecture B. Let D be a division ring and M a nilpotent maximal subgroup of D∗.Then D is commutative. The special case of Conjecture B makes the following interesting conjecture. Conjecture C. Let D be a division ring and M an abelian maximal subgroup of D∗.Then D is commutative. The present article contains three sections. In Section 1 we establish that C∗ ∪ C∗j(C is the field of complex numbers) is a solvable maximal subgroup of H∗,whereH denotes real quaternions. Thus this gives a counterexample to Conjecture A. Also in Section 1 we show that if M is a nilpotent maximal subgroup of D∗,whereD is a division ring and M\F contains an algebraic element over F ,thenM is an abelian group. In Section 2 we construct a family of maximal subgroups of GLn(D), n>1, and completely classify the reducible maximal subgroups of GLn(D). Moreover, Conjecture C will be extended to skew linear groups and it will be shown that if D is an infinite- dimensional division ring over its center F and GLn(D) is algebraic over F ,thenGLn(D) contains no abelian maximal subgroup. Finally, in Section 3 we generalize our results to general skew linear groups and prove that for a locally finite-dimensional division ring D, if M is a locally nilpotent maximal subgroup of GLn(D),thenM is abelian. Also we show that locally finiteness of a maximal subgroup of GLn(D) implies that GLn(D) is locally finite. Before stating our results, we fix some notations. Throughout, D is a division ring with center F , we shall denote their multiplicative groups by D∗, F ∗, respectively. For any group G and any subset X of G, CG(X),andNG(X) denote respectively the centralizer and the normalizer of X in G. In this article Gr denotes the rth derived subgroup of G.The set of primes which occur as the order of some element of G will be denoted by Π(G). Throughout this paper H<Gmeans that H is a not necessarily proper subgroup of G. By a skew linear group we mean a subgroup of the general skew linear group GLn(D),for n some division ring D.LetG be a subgroup of GLn(D) and set V = D , the space of row n-vectors over D.ThenV is a D-G bimodule in the obvious way. We say that G is an irreducible (respectively reducible, completely reducible) subgroup of GLn(D), whenever V is irreducible (respectively reducible, completely reducible) as a D-G bimodule. Also G is said to be absolutely irreducible if F [G]=Mn(D).ForafieldF , we call a subgroup S. Akbari et al. / Journal of Algebra 259 (2003) 201–225 203 G of GLn(F ) nonmodular if G is torsion and Char F/∈ Π(G). A field extension K/F is said to be radical if for every x ∈ K there exists a natural number n(x) such that xn(x) ∈ F . For a subset X of Mn(D) and a subring R of D, R[X] denotes the subalgebra generated by R and X.AlsoifX is a subset of D,thenR(X) denotes the subdivision ring generated by R and X. We write [V : K]r for the dimension of a right vector space V over a field K. For a given ring R, the group of units of R is denoted by U(R). 1. Maximal subgroups of division rings Division rings have a more complicated structure than fields and much less is known about them. It is easily seen that the multiplicative group of a field does not necessarily have a maximal subgroup. For example, if M is a maximal subgroup of index n of C∗, then for any x ∈ C∗, xn is contained in M, implying M = C∗, since indeed any element of C∗ is an nth power. It seems that the multiplicative group of a noncommutative division ring has maximal subgroup and it was conjectured first in [2]. The structure of a maximal subgroup of a group forces some properties in the group. For example, in [17] there is a theorem which says that if a finite group contains an abelian maximal subgroup, then it should be solvable. It is not true in general case, because there are some examples of nonsolvable groups which have finite abelian maximal subgroups, see [15]. Since GLn(D) is violently non-abelian it is intuitively see that GLn(D) has no abelian maximal subgroup. It is a natural question to ask what can be said about the structure of division rings whose multiplicative subgroups have an abelian maximal group or a solvable maximal subgroup. In what follows we shall give a counterexample to Conjecture A. Let us record the following technical lemma. Lemma 1. H∗ = C∗, j + 1 . Proof. Let G = C∗, j + 1 . We claim that if w,z ∈ C with |z|=|w| and z + j ∈ G then w + j ∈ G.Forz = 0 it is clear, so assume that z = 0. Since |z|=|w|, there exists some element t ∈ C such that wz−1 = t/t¯.Now,(z + j)t = t(w¯ + j) ∈ G.So,w + j ∈ G which proves the claim. Since each element of H is of the form z1 + z2jwherez1,z2 ∈ C, to prove the lemma it suffices to show that for any a ∈ C we have a + j ∈ G.Forr>0, let 1 − r 1 − r 2 u = + 1 − i. 1 + r 1 + r So |u|=1 and using our claim, since 1+j ∈ G,wehaveu+j ∈ G. Thus (1+j)(u+j) ∈ G, which implies that (u − 1)/(u¯ + 1) + j ∈ G. On the other hand, we have √ u − 1 = r. u¯ + 1 Since r>0 was arbitrary, by our claim for all a ∈ C we have a + j ∈ G. ✷ 204 S. Akbari et al. / Journal of Algebra 259 (2003) 201–225 The following theorem asserts a very interesting property of real quaternions, that there is just one subgroup between C∗ and H∗; actually it gives a counterexample to Conjecture 3 of [3]. Theorem 1. Let M be a proper subgroup of H∗ containing C∗ then M = C∗ or M = C∗ ∪ C∗j. Therefore H∗ has a solvable maximal subgroup. ∗ ∗ ∗ ∗ Proof. It suffices to prove that for any z0 ∈ C , H = C ,z0 + j .LetG = C ,z0 + j | |2 = where z0 r0 √> 0. By a similar√ method used in the proof of the previous lemma, since z0 + j ∈ G, both r0 i + jand r0 + jareinG.So, √ √ √ √ r0 + j r0 i + j = (r0i − 1) + r0 − r0 i j ∈ G. Let us put r0i − 1 z1 = √ √ . r0 − r0 i | |2 = 2 + + ∈ | |2 = Then√ z1 √ (r0 1)/(2r0) 1andz1 j G.