Sharif Digital Repository

Let D be an F-central division algebra of index n. Here we present a criterion for the triviality of the group G(D) = D*/Nrd D/F(D*)D′ and thus generalizing various related results published recently. To be more precise, it is shown that G(D) = 1 if and only if SK 1(D) = 1 and F *2 = F *2n. Using this, we investigate the role of some particular subgroups of D* in the algebraic structure of D. In this direction, it is proved that a division algebra D of prime index is a symbol algebra if and only if D* contains a non-abelian nilpotent subgroup. More applications of this criterion including the computation of G(D) and the structure of maximal subgroups of D* are also investigated  

Given a divisible finite field extension K/F, the structure of Br(F), the Brauer group of F, is investigated. It is shown that, if F is indivisible, then Br(F) ≅ ℤ 2, which generalizes the Frobenius Theorem. As a consequence, when F is indivisible, the class of all finite dimensional non-commutative F-central division algebras D having radicable multiplicative groups D* is determined. In fact, it is proved that the following statements are equivalent: (1) D is radicable, (2) D contains a divisible subfield K/F, and (3) D is the ordinary quaternion division algebra and F(√-1) is divisible  

Let F be a field and M be a maximal subgroup of the multiplicative group F* = F {0} of index p. It is proved that if M is divisible, then Br(F)p ≠ 0 if and only if p = 2 and F is Euclidean. Furthermore, it is shown that in this case F* contains a divisible maximal subgroup if and only if F* is isomorphic to the multiplicative group of a real closed field. © 2009 Iranian Mathematical Society  

Let D be an infinite division algebra of finite dimension over its centre Z(D) = F, and n a positive integer. The structure of maximal subgroups of skew linear groups are investigated. In particular, assume N is a normal subgroup of GLn(D) and M is a maximal subgroup of N containing Z(N). It is shown that if M/Z(N) is finite, then N is central. © Inst. Math. CAS 2002  

Let D be a noncommutative division algebra of finite dimension over its centre F. Given a maximal subgroup M of GLn (D) with n ≥ 1, it is proved that either M contains a noncyclic free subgroup or there exists a finite family {Ki}1r of fields properly containing F with Ki* ⊂ M for all 1 ≤ i ≤ r such that M/A is finite if Char F = 0 and M/A is locally finite if Char F = p > 0, where A = K1* x ⋯ x Kr*. © 2004 Elsevier Inc. All rights reserved  

Let D be a division algebra of finite dimension over its center F. Given a noncommutative maximal subgroup M of D*:= GL1(D), it is proved that either M contains a noncyclic free subgroup or there exists a maximal subfield K of D which is Galois over F such that K* is normal in M and M/K*≅Gal(K/F). Using this result, it is shown in particular that if D is a noncrossed product division algebra, then M does not satisfy any group identity. © 2001 Academic Press  

Let D be a finite-dimensional F-central division algebra. A criterion is given for D to be a supersoluble (nilpotent) crossed product division algebra in terms of subgroups of the multiplicative group D* of D. More precisely, it is shown that D is a supersoluble (nilpotent) crossed product if and only if D* contains an abelian-by-supersoluble (abelian-by-nilpotent) generating subgroup  

Let D be an infinite division algebra of finite dimension over its center. Assume that N is a subnormal subgroup of GLn(D) with n≥1. It is shown that if N is finitely generated, then N is central. © 2000 Academic Press  

Given a non-commutative finite dimensional F-central division algebra D, we study conditions under which every non-abelian maximal subgroup M of GL n(D) contains a non-cyclic free subgroup. In general, it is shown that either M contains a non-cyclic free subgroup or there exists a unique maximal subfield K of Mn(D) such that NGLn<(D) (K*)=M, K* M, K/F is Galois with Gal(K/F) ≅ M/K*, and F[M]=Mn(D). In particular, when F is global or local, it is proved that if ([D:F],Char(F))=1, then every non-abelian maximal subgroup of GL 1(D) contains a non-cyclic free subgroup. Furthermore, it is also shown that GLn(F) contains no solvable maximal subgroups provided that F is local or global and n ≥ 5. ©... 

Let D be an F-central non-commutative division ring. Here, it is proved that if GL n(D) contains a non-abelian soluble maximal subgroup, then n = 1, [D : F] < ∞, and D is cyclic of degree p, a prime. Furthermore, a classification of soluble maximal subgroups of GL n(F) for an algebraically closed or real closed field F is also presented. We then determine all soluble maximal subgroups of GL 2(F) for fields F with Char F ≠ 2  

Let D be a finite dimensional F-central division algebra and G an irreducible subgroup of D*: = GL1(D). Here we investigate the structure of D under various group identities on G. In particular, it is shown that when [D : F] = p2, p a prime, then D is cyclic if and only if D* contains a nonabelian subgroup satisfying a group identity  

Let D be a division algebra of prime degree p. A set of criteria is given for cyclicity of D in terms of subgroups of the multiplicative group D* of D. It is essentially shown that D is cyclic if and only if D* contains a nonabelian metabelian subgroup