Free Subgroups in Maximal Subgroups of GL1(D)

Mahdavi-Hezavehi

2005

Algebra Colloq.

Let D be a division ring with centre F . Assume that M is a maximal subgroup of GL n (D), n ≥ 1 such that Z(M ) is algebraic over F . Group identities on M and polynomial identities on the F -linear hullWhen D is noncommutative 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 GL n (D) and M is a maximal subgroup of N .If M satisfies a group identity, it is shown that M is abelian-by-finite.

mentioning

confidence: 99%

Identities on Maximal Subgroups of GL_n(D)

Mahdavi-Hezavehi

2005

Algebra Colloq.

“…Thus, by Tits alternative, we know that G 1 is soluble-by-finite, i.e., there is a soluble normal subgroup N of G 1 such that G 1 /N is finite. Now, by [4,Lemma 3], N is abelian-by-finite. Thus, G 1 is abelian-by-finite.…”

Section: Corollarymentioning

confidence: 98%

Crossed product conditions for division algebras of prime power degree

Mahdavi-Hezavehi

2005

Journal of Algebra

Let D be an F -central division algebra of degree p r , p a prime. A set of criteria is given for D to be a crossed product in terms of irreducible soluble or abelian-by-finite subgroups of the multiplicative group D * of D. Using the Amitsur's classification of finite subgroups of D * and the Tits alternative, it is shown that D is a crossed product if and only if D * contains an irreducible soluble subgroup. Further criteria are also presented in terms of irreducible abelian-by-finite subgroups and irreducible subgroups satisfying a group identity. Using the above results, it is shown that if D * contains an irreducible finite subgroup, then D is a crossed product.

“…Our next observation is about maximal subgroups of skew linear groups; these groups have been studied in a series of papers, see, e.g., [1,7,16,17]. In [7], it was shown that if D is an infinite division ring and m is a natural number, then every nilpotent maximal subgroup of GL m .D/ is abelian.…”

Section: Introductionmentioning

confidence: 99%

Some skew linear groups with Engel's condition

Ramezan-Nassab

Journal of Group Theory

2012

Abstract. Let D be a non-commutative division ring, m and n two natural numbers, M a maximal and N a subnormal subgroup of GL m .D/. In this paper, among other results, we show that: (1) if N is an n-Engel group, then it is central; (2) if m > 1 and M is locally nilpotent, then M is abelian; (3) if m > 1 and M is n-Engel, then the Hirsch-Plotkin radical of M is abelian. Also, we define some generalized Engel conditions on groups, and then we prove similar results, as quoted, for these groups.