Examples of using Linear groups in English and their translations into Danish
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In particular he studied linear groups and groups of automorphic functions.
However we should note that before publishing the book, Dickson had already published 43 research papers in the preceding five years which, with the exception of seven,were all on finite linear groups.
Dickson published 17 books in addition to Linear groups with an exposition of the Galois field theory.
Among these results we mention the local theorem for the class of groups representable by matrices of a given order, andalso the theorem on residual finiteness of finitely generated linear groups.
In 1901 his famous book Linear groups with an exposition of the Galois field theory was published.
His work in group theory continued andhe proved important results on linear groups and, in particular, on linear soluble groups. .
Loewy worked on linear groups, the algebraic theory of differential equations and actuarial mathematics.
In 1955 he produced two important papers on finite simple groups, proving that the only coincidences in orders of the known(in 1955)finite simple groups were those given by Dickson in his Linear groups.
He produced results in invariant theory, linear groups, Lie groups and generalised some of Emmy Noether 's results on rings.
In the area of transform theory he worked on the representation and uniqueness of integral transforms, on approximation, andon linear transformations which satisfy functional relations arising from representations of linear groups.
In 1931 he gave a course on linear groups of transformations on Hilbert space when invited to give the prestigious Peccot Foundation lectures at the College of France.
The book was a revised and expanded version of his 1896 doctoral thesis. However we should note that before publishing the book, Dickson had already published 43 research papers in the preceding five years which, with the exception of seven,were all on finite linear groups.
Dickson published 17 books in addition to Linear groups with an exposition of the Galois field theory. The 3-volume History of the Theory of Numbers(1919-23) is another famous work still much consulted today.
In and his work on representations of infinite groups by matrices is commented on: Among these results we mention the local theorem for the class of groups representable by matrices of a given order, andalso the theorem on residual finiteness of finitely generated linear groups.
The book here announced proposes to treat of linear congruence groups, or more generally, of linear groups in a Galois field, a subject enriched by the labors of Galois, Betti, Mathieu[ Emile Mathieu], Jordan and many recent writers.
Dickson presented a unified, complete, andgeneral theory of the classical linear groups- not merely over the prime field GF(p) as Jordan had done- but over the general finite field GF(pn), and he did this against the backdrop of a well-developed theory of these underlying fields….
In the proposal for his book, sent to Klein, Dickson wrote:The book here announced proposes to treat of linear congruence groups, or more generally, of linear groups in a Galois field, a subject enriched by the labors of Galois, Betti, Mathieu[ Emile Mathieu], Jordan and many recent writers.
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The group he studied was the special linear group of 2 by 2 matrices over the ring of integers modulo p n.
In 1901 Schur obtained his doctorate with a thesis which examined rational representations of the general linear group over the complex field.
Generalising a result of Fuchs on linear differential equations,Jordan was led to study the finite subgroups of the general linear group of n n matrices over the complex numbers.
A second major piece of work on finite groups was the study of the general linear group over the field with p elements, p prime.
Schur returned to work on representation theory with renewed vigour and he was able to complete the programme of research begun in his doctoral dissertation andgive a complete description of the rational representations of the general linear group.
Dickson received a Ph.D. from the University of Chicago in 1896 for a dissertation entitled The Analytic Representation of Substitutions on a Power of a Prime Number of Letters with a Discussion of the Linear Group.
They discussed the simple Lie algebras which they knew about and Killing conjectured(wrongly)on 12 April 1886 that the only simple algebras were those related to the special linear group and orthogonal groups. .
Another generalisation, this time of work by Hermite on quadratic forms with integral coefficients,led Jordan to consider the special linear group of n n matrices of determinant 1 over the complex numbers acting of the vector space of complex polynomials in n indeterminates of degree m.
The group he studied was the special linear group of 2 by 2 matrices over the ring of integers modulo pn. Schur had solved the problem for the case n 1, where the matrices are over a prime field, and the case of n 2 had been solved in the 1930s.
At Chicago he was supervised by Eliakim Moore, but others there influenced him, for example Bolza and Maschke.Dickson received a Ph.D. from the University of Chicago in 1896 for a dissertation entitled The Analytic Representation of Substitutions on a Power of a Prime Number of Letters with a Discussion of the Linear Group.
The group he studied was the special linear group of 2 by 2 matrices over the ring of integers modulo pn. Schur had solved the problem for the case n 1, where the matrices are over a prime field, and the case of n 2 had been solved in the 1930s. Kloosterman solved the general case in two papers The behaviour of general theta functions under the modular group and the characters of binary modular congruence groups which occupy 130 pages of the Annals of Mathematics in 1946.
Around 1914 Schur's interest in representations of groups was put to one side while he worked on other topics but, around 1925, developments in theoretical physics showed that group representations were of fundamental importance in that subject. Schur returned to work on representation theory with renewed vigour and he was able to complete the programme of research begun in his doctoral dissertation andgive a complete description of the rational representations of the general linear group.