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In the proof of Subcase 2 of Theorem 5 in Mahdavi-Hezavehi (2004) [2], not all the required steps are considered properly. Here we shall deal with the remaining step of Subcase 2. Therefore, this completes the proof of the main result that if D is an F-central finite-dimensional division algebra and M is a maximal subgroup of GLn (D), D ≠ F, n > 1, and M / M ∩ F* is torsion, then M is abelian-by-finite. © 2009 Elsevier Inc. All rights reserved  

Let D be a noncommutative finite dimensional F-central division algebra, and let N be a normal subgroup of GLn(D) with n≥1. Given a maximal subgroup M of N, it is proved that either M contains a noncyclic free subgroup, or there exists an abelian subgroup A and a finite family {Ki}1 r of fields properly containing F with Ki* ∩ N ⊃ 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* ×...× Kr*  

Let D be a division ring with centre F. Assume that M is a maximal subgroup of GLn(D) (n ≥ 1) such that Z(M) is algebraic over F. Group identities on M and polynomial identities on the F-linear hull F[M] are investigated. It is shown that if F[M] is a PI-algebra, then [D : F] < ∞. When D is non-commutative and F is infinite, it is also proved that if M satisfies a group identity and F[M] is algebraic over F, then we have either M = K* where K is a field and [D : F] < ∞, or M is absolutely irreducible. For a finite dimensional division algebra D, assume that N is a subnormal subgroup of GLn(D) and M is a maximal subgroup of N. If M satisfies a group identity, it is shown that M is... 

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. ©... 

Given an F-central simple algebra A = M n(D), denote by A′ the derived group of its unit group A*. Here, the Frattini subgroup Φ(A*) of A* for various fields F is investigated. For global fields, it is proved that when F is a real global field, then Φ(A*) = Φ(F*)Z(A′), otherwise Φ(A*) = ∩ pdoes not dividedeg(A) F *p. Furthermore, it is also shown that Φ(A*) = k* whenever F is either a field of rational functions over a divisible field k or a finitely generated extension of an algebraically closed field k  

Given a division ring D with center F, the structure of maximal subgroups M of GL n(D) is investigated. Suppose D ≠ F or n > 1. It is shown that if M/(M ∩ F*) is locally finite, then char F = p > 0 and either n = 1, [D:F] = p 2 and M ∪ {0} is a maximal subfield of D, or D = F, n = p, and M ∪ {0} is a maximal subfield of M p(F), or D = F and F is locally finite. It is also proved that the same conclusion holds if M/(M ∩ F*) is torsion and D is of finite dimension over F. Furthermore, it is shown that if the r-th derived group M (r) of M is locally finite, then either M (r) is abelian or F is algebraic over its prime subfield  

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  

Given a finite dimensional F-central simple algebra A = Mn(D), the connection between the Frattini subgroup Φ(A*) and Φ(F*) via Z(A′), the center of the derived group of A*, is investigated. Setting G = F* ∩ Φ(A*), it is shown that Φ (F*)Z(A′) ⊂ G ⊂ (∩p F*p) Z(A′) where the intersection is taken over primes p not dividing the degree of A. Furthermore, when F is a local or global field, the group G is completely determined. Using the above connection, Φ(A*) is also calculated for some particular division rings D  

Let D be a noncommutative finite dimensional F-central division algebra and M a noncommutative maximal subgroup of GLn(D). It is shown that either M contains a noncyclic free subgroup or M is absolutely irreducible and there exists a unique maximal subfield K of Mn(D) such that K*M, K∕F is Galois with Gal(K∕F)≅M∕K* and Gal(K∕F) is a finite simple group. © 2017 Taylor & Francis  

Let D be an F-central division algebra of index n. Here we investigate a conjecture posed in [R. Hazrat et al., Reduced K-theory and the group G(D) = D*/F*D′, in: Algebraic K-theory and its Applications, Trieste, 1997, pp. 403-409] that if D is not a quaternion algebra, then the group G0(D) = D*/F*D′ is non-trivial. Assume that either D is cyclic or F contains a primitive pth root of unity for some prime p n. Using Merkurjev-Suslin Theorem, it is essentially shown that if none of the primary components of D is a quaternion algebra, then G(D) = D*/RND/F(D*) D′ ≠ 1. In this direction, we also study a conjecture posed in [S. Akbari, M. Mahdavi-Hezavehi, J. Pure Appl. Algebra 171 (2002) 123-131]... 

Let R be a local ring, with maximal ideal m, and residue class division ring R/m = D. Denote by R* = GL1(R), the group of units of R. Here we investigate some algebraic structure of subnormal and maximal subgroups of R*. For instance, when D is of finite dimension over its center, it is shown that finitely generated subnormal subgroups of R* are central. It is also proved that maximal subgroups of R* are not finitely generated. Furthermore, assume that P is a nonabelian maximal subgroup of R* such that P contains a noncentral soluble normal subgroup of finite index, it is shown that D is a crossed product division algebra. Copyright © 2004 by Marcel Dekker, Inc  

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  

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*, then D is commutative. In connection with this conjecture we show that if F[M]F contains an algebraic element over F, then M 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* then... 

Recently, it is shown that, if D is a finite-dimensional division ring, then GLn(D) is not finitely generated. Our object here is to provide a general framework for the groups of units of left Artinian rings. We prove that, if R is an infinite F-algebra of finite dimension over F, then U(R) is not finitely generated. We show that any infinite subnormal subgroup of GL n (D) has no finite maximal subgroup. Also, we prove that for any infinite left Artinian ring R, U(R) has no finite maximal subgroup, which is a result analogous to that for rings. © Inst. Math. CAS 2002