Conjugacy and factorization results on matrix groups

Thomas Laffey

Banach Center Publications (1994)

  • Volume: 30, Issue: 1, page 203-221
  • ISSN: 0137-6934

Abstract

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In this survey paper, we present (mainly without proof) a number of results on conjugacy and factorization in general linear groups over fields and commutative rings. We also present the additive analogue in matrix rings of some of these results. The first section deals with the question of expressing elements in the commutator subgroup of the general linear group over a field as (simple) commutators. In Section 2, the same kind of problem is discussed for the general linear group over a commutative ring. In Section 3, the analogous question for additive commutators is discussed. The case of integer matrices is given special emphasis as this is an area of current interest. In Section 4, factorizations of an element A ∈ GL(n,F) (F a field) in which at least one of the factors preserves some form (e.g. is symmetric or skew-symmetric) is considered. An application to the size of abelian subgroups of finite p-groups is presented. In Section 5, a curious interplay between additive and multiplicative commutators in M n ( F ) (F a field) is identified for matrices of small size and a general factorization theorem for a polynomial using conjugates of its companion matrix is presented.

How to cite

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Laffey, Thomas. "Conjugacy and factorization results on matrix groups." Banach Center Publications 30.1 (1994): 203-221. <http://eudml.org/doc/262781>.

@article{Laffey1994,
abstract = {In this survey paper, we present (mainly without proof) a number of results on conjugacy and factorization in general linear groups over fields and commutative rings. We also present the additive analogue in matrix rings of some of these results. The first section deals with the question of expressing elements in the commutator subgroup of the general linear group over a field as (simple) commutators. In Section 2, the same kind of problem is discussed for the general linear group over a commutative ring. In Section 3, the analogous question for additive commutators is discussed. The case of integer matrices is given special emphasis as this is an area of current interest. In Section 4, factorizations of an element A ∈ GL(n,F) (F a field) in which at least one of the factors preserves some form (e.g. is symmetric or skew-symmetric) is considered. An application to the size of abelian subgroups of finite p-groups is presented. In Section 5, a curious interplay between additive and multiplicative commutators in $M_n(F)$ (F a field) is identified for matrices of small size and a general factorization theorem for a polynomial using conjugates of its companion matrix is presented.},
author = {Laffey, Thomas},
journal = {Banach Center Publications},
keywords = {survey paper; conjugacy; factorization; linear groups; commutative rings; matrix rings; commutator subgroup; additive commutators; multiplicative commutators},
language = {eng},
number = {1},
pages = {203-221},
title = {Conjugacy and factorization results on matrix groups},
url = {http://eudml.org/doc/262781},
volume = {30},
year = {1994},
}

TY - JOUR
AU - Laffey, Thomas
TI - Conjugacy and factorization results on matrix groups
JO - Banach Center Publications
PY - 1994
VL - 30
IS - 1
SP - 203
EP - 221
AB - In this survey paper, we present (mainly without proof) a number of results on conjugacy and factorization in general linear groups over fields and commutative rings. We also present the additive analogue in matrix rings of some of these results. The first section deals with the question of expressing elements in the commutator subgroup of the general linear group over a field as (simple) commutators. In Section 2, the same kind of problem is discussed for the general linear group over a commutative ring. In Section 3, the analogous question for additive commutators is discussed. The case of integer matrices is given special emphasis as this is an area of current interest. In Section 4, factorizations of an element A ∈ GL(n,F) (F a field) in which at least one of the factors preserves some form (e.g. is symmetric or skew-symmetric) is considered. An application to the size of abelian subgroups of finite p-groups is presented. In Section 5, a curious interplay between additive and multiplicative commutators in $M_n(F)$ (F a field) is identified for matrices of small size and a general factorization theorem for a polynomial using conjugates of its companion matrix is presented.
LA - eng
KW - survey paper; conjugacy; factorization; linear groups; commutative rings; matrix rings; commutator subgroup; additive commutators; multiplicative commutators
UR - http://eudml.org/doc/262781
ER -

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