Crossed product conditions for central simple algebras in terms of irreducible subgroups

Keshavarzipour, T.; Mahdavi-Hezavehi, M.

doi:10.1016/j.jalgebra.2007.05.002

Cited by 7 publications

(2 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…From Lemma 2.18, it follows that D is a crossed product of Here, we should point out that the above results concerning the structure of soluble-byfinite subgroups of division algebras were nicely obtained by Wehrfritz for the soluble-byfinite subgroups of GL n (D), where D is a division algebra in [107]. His results demonstrates that if G is a soluble-by-finite subgroup of GL n (D), then G has an abelian normal subgroup of finite index dividing b(n) deg(D) 2n , where b(n) is an integer valued function which depends only on n. By an interesting result of [45], the existence of a finite absolutely irreducible subgroup in a division algebra D guarantees that D is a crossed product division algebra. Here, we give a very short proof for this fact.…”

Section: Soluble-by-finite Subgroups Of Division Algebrasmentioning

confidence: 92%

“…The first significant result concerning Question 2.5, was obtained in [60] where the author showed that if G is a soluble maximal subgroup (this means that G is a maximal subgroup of D * that is soluble), then the question has a positive answer. Afterwards, this question became the subject of a series of papers including [13], [14], [47] and [45]. In [13], it was proved that if G is soluble, then D is a quasi-crossed product division algebra in the sense that it contains a tower of subfields F = Z(D) K ⊆ L such that K/F is Galois, L is a maximal subfield and L/K is an abelian Galois extension.…”

Section: Theorem 24 ([69]mentioning

confidence: 99%

See 1 more Smart Citation

Multiplicative groups of division rings

Hazrat¹,

Mahdavi-Hezavehi²,

Motiee³

2014

Mathematical Proceedings of the Royal Irish Academy

View full text Add to dashboard Cite

Abstract. Exactly 170 years ago, the construction of the real quaternion algebra by William Hamilton was announced in the Proceedings of the Royal Irish Academy. It became the first example of non-commutative division rings and a major turning point of algebra. To this day, the multiplicative group structure of quaternion algebras have not completely been understood. This article is a long survey of the recent developments on the multiplicative group structure of division rings.

show abstract

Section: Soluble-by-finite Subgroups Of Division Algebrasmentioning

confidence: 92%

Section: Theorem 24 ([69]mentioning

confidence: 99%

Multiplicative groups of division rings

Hazrat¹,

Mahdavi-Hezavehi²,

Motiee³

2014

Mathematical Proceedings of the Royal Irish Academy

View full text Add to dashboard Cite

show abstract

LOCALLY NILPOTENT SUBGROUPS OF $\mathrm {GL}_n(D)$

FALLAH-MOGHADDAM,

DORBIDI

2024

Bull. Aust. Math. Soc.

View full text Add to dashboard Cite

Let A be an F-central simple algebra of degree $m^2=\prod _{i=1}^k p_i^{2\alpha _i}$ and G be a subgroup of the unit group of A such that $F[G]=A$ . We prove that if G is a central product of two of its subgroups M and N, then $F[M]\otimes _F F[N]\cong F[G]$ . Also, if G is locally nilpotent, then G is a central product of subgroups $H_i$ , where $[F[H_i]:F]=p_i^{2\alpha _i}$ , $A=F[G]\cong F[H_1]\otimes _F \cdots \otimes _F F[H_k]$ and $H_i/Z(G)$ is the Sylow $p_i$ -subgroup of $G/Z(G)$ for each i with $1\leq i\leq k$ . Additionally, there is an element of order $p_i$ in F for each i with $1\leq i\leq k$ .

show abstract

Solvable Crossed Product Algebras Revisited

Brown¹,

Pumpluen²

2019

Glasgow Math. J.

View full text Add to dashboard Cite

For any central simple algebra over a field F which contains a maximal subfield M with non-trivial automorphism group G = AutF (M ), G is solvable if and only if the algebra contains a finite chain of subalgebras which are generalized cyclic algebras over their centers (field extensions of F ) satisfying certain conditions. These subalgebras are related to a normal subseries of G. A crossed product algebra F is hence solvable if and only if it can be constructed out of such a finite chain of subalgebras. This result was stated for division crossed product algebras by Petit, and overlaps with a similar result by Albert which, however, is not explicitly stated in these terms. In particular, every solvable crossed product division algebra is a generalized cyclic algebra over F .

show abstract

Crossed product conditions for central simple algebras in terms of irreducible subgroups

Cited by 7 publications

References 11 publications

Multiplicative groups of division rings

Multiplicative groups of division rings

LOCALLY NILPOTENT SUBGROUPS OF $\mathrm {GL}_n(D)$

Solvable Crossed Product Algebras Revisited

Contact Info

Product

Resources

About