Conjugacy and factorization results on matrix groups (original) (raw)
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It is known that that the centralizer of a matrix over a finite field depends, up to conjugacy, only on the type of the matrix, in the sense defined by J. A. Green. In this paper an analogue of the type invariant is defined that in general captures more information; using this invariant the result on centralizers is extended to arbitrary fields. The converse is also proved: thus two matrices have conjugate centralizers if and only if they have the same generalized type. The paper ends with the analogous results for symmetric and alternating groups.
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Let A be a groupoid (a set with a single binary operation). Let k≥2 and σ be a permutation of the set {1,⋯,k}. The following binary operation is considered on the set A K : x∘ σy=(x 1 ,⋯,x k )∘ σ(y 1 ,⋯,y k )=(x 1 σ(y 1 ),⋯,x k σ(y k )) so as an l-ary operation for l≥2: [x 1 ,x 2 ,⋯,x l ] l,σ,k =x 1 ∘ σ(x 2 ∘ σ(⋯(x l-2 ∘ σ(x l-1 ∘ σx l ))⋯))· Let A be a group and let σ satisfy the condition σ l =σ. Then (A k ,[] l,σ,k ) is an l-ary group. Let in addition B be a normal subgroup of A such that the factor group A/B is cyclic and has order a divisor of l-1. In this case for any element H of the factor group A/B, the Cartesian power H k is closed relatively to the l-ary operation [] l,σ,k . The authors study the properties of this l-ary operation on Cartesian powers of conjugate group classes of the group A associated to B.
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Group rings, matrix rings, and polynomial identities
Transactions of The American Mathematical Society, 1972
This paper studies the question, if R is a ring satisfying a polynomial identity, what polynomial identities are satisfied by group rings and matrix rings over R? Theorem 2.6. If R is an algebra over a field with at least q elements, and R satisfies x^ = 0, and G is a group with an abelian subgroup of index k, then the group ring R(G) satisfies x = 0, where t = qk +2. Theorem 3.2. If R is a ring satisfying a standard identity, and G is a finite group, then A?(G) satisfies a standard identity. Theorem 3.4. If R is an algebra over a field, and R satisfies a standard identity, then the k-by-k matrix ring R, satisfies a standard identity. Each theorem specifies the degree of the polynomial identity.
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We investigate the commutators of elements of the group UT(∞, R) of infinite unitriangular matrices over an associative ring R with 1 and a commutative group R * of invertible elements. We prove that every unitriangular matrix of a specified form is a commutator of two other unitriangular matrices. As a direct consequence we give a complete characterization of the lower central series of the group UT(∞, R) including the width of its terms with respect to basic commutators and Engel words. With an additional restriction on the ring R, we show that the derived subgroup of T(∞, R) coincides with the group UT(∞, R). The obtained results generalize the results obtained for triangular groups over a field.