The General Linear Group Over a Ring

Han, Juncheol

doi:10.4134/bkms.2006.43.3.619

Cited by 19 publications

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“…Using Shoda's result, or otherwise, we can prove that Aut C n p r ∼ = GL(n, Z p r ). The following lemma of Han [3,Lemma 2.8] gives the order of this group, which also follows from [8]. Since there are no common direct factors among A and the H βi i (1 < i ≤ n), Theorem 2.2 gives that Aut G is isomorphic with the set of (n + 1) × (n + 1) matrices B = (B i,j ), where …”

Section: General Direct Productsmentioning

confidence: 99%

“…Then: |Aut C 3 2 | = 168, |Aut C 2 3 | = 48, |Aut S 3 | 2 = 36, |Aut Q 8 | 2 = 576, |Hom (C 2 3 , C 3 2 )| = 1, |Hom (S 3 , C 3 2 )| 2 = 64, |Hom (Q 8 , C 3 2 )| 2 = 2 12 , |Hom (C 3 2 , C 2 3 )| = |Hom (S 3 , C 2 3 )| 2 = |Hom (Q 8 , C 2 3 )| 2 = 1, |Hom (C 3 2 , Z(S 3 ))| 2 = |Hom (C 2 3 , Z(S 3 ))| 2 = |Hom (Q 8 , Z(S 3 ))| 4 = 1, |Hom (C 3 2 , Z(Q 8 ))| 2 = 64, |Hom (C 2 3 , Z(Q 8 ))| 2 = 1, |Hom (S 3 , Z(Q 8 ))| 4 = 16, |Hom (S 3 , Z(S(3)))| 2 = 1 and |Hom (Q 8 , Z(Q 8 ))| 2 = 16. …”

Section: Arch Mathmentioning

confidence: 99%

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Automorphisms of direct products of finite groups II

Bidwell

2008

Arch. Math.

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In this paper we first find the automorphism group of the direct product of n copies of an indecomposable non-abelian group. We describe the automorphism group as matrices with entries which are homomorphisms between the n direct factors. We then use this description with a generalization of a result by Bidwell, Curran, and McCaughan on Aut (H ×K), where H and K have no common direct factor, to provide structure and order theorems for an arbitrary direct product.

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Section: General Direct Productsmentioning

confidence: 99%

Section: Arch Mathmentioning

confidence: 99%

Automorphisms of direct products of finite groups II

Bidwell

2008

Arch. Math.

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“…Proof. The result is an easy consequence of the Chinese Remainder Theorem (see also [Han06]). 2 Proposition 2.3.…”

Section: Principal Congruence Subgroupsmentioning

confidence: 80%

Cohomology of Braids, Principal Congruence Subgroups and Geometric Representations

Callegaro¹,

Cohen²,

Salvetti³

2014

The Quarterly Journal of Mathematics

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Abstract. The main purpose of this article is to give the integral cohomology of classical principal congruence subgroups in SL(2, Z) as well as their analogues in the third braid group with local coefficients in symmetric powers of the natural symplectic representation. The resulting answers (1) correspond to certain modular forms in characteristic zero, and (2) the cohomology of certain spaces in homotopy theory in characteristic p. The torsion is given in terms of the structure of a "p-divided power algebra".The work is an extension of the work in [CCS12] as well as extensions of a classical computation of Shimura to integral coefficients. The results here contrast the local coefficients such as that in [Loo96] and [Til10].

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“…Some important ring-theoretic concepts, such as coprime and maximal ideals, idempotents, local rings, semilocal rings and finite chain rings, are also introduced here. For a more thorough treatment of these topics, the reader is referred to [2,3,4].…”

Section: Preliminaries and Definitionsmentioning

confidence: 99%

One-Sided $k$-Orthogonal Matrices Over Finite Semi-Local Rings And Their Codes

Sison,

Repizo

2021

Preprint

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Let R be a finite commutative ring with unity 1 R and k ∈ R. Properties of one-sided k-orthogonal n × n matrices over R are presented. When k is idempotent, these matrices form a semigroup structure. Consequently new families of matrix semigroups over certain finite semi-local rings are constructed. When k = 1 R , the classical orthogonal group of degree n is obtained. It is proved that, if R is a semi-local ring, then these semigroups are isomorphic to a finite product of korthogonal semigroups over fields. Finally, the antiorthogonal and self-orthogonal matrices that give rise to leading-systematic self-dual or weakly self-dual linear codes are discussed.

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The General Linear Group Over a Ring

Cited by 19 publications

References 0 publications

Automorphisms of direct products of finite groups II

Automorphisms of direct products of finite groups II

Cohomology of Braids, Principal Congruence Subgroups and Geometric Representations

One-Sided $k$-Orthogonal Matrices Over Finite Semi-Local Rings And Their Codes

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