Auf VektorrĂ€umen ĂŒber â oder â fĂŒhren wir einen LĂ€ngenbegriff ein, eine Norm. Dies fĂŒhrt zum Grenzwertbegriff auf solchen VektorrĂ€umen. Der Normbegriff ist noch sehr allgemein. Neben dem LĂ€ngenbegriff der âŠ
Auf VektorrĂ€umen ĂŒber â oder â fĂŒhren wir einen LĂ€ngenbegriff ein, eine Norm. Dies fĂŒhrt zum Grenzwertbegriff auf solchen VektorrĂ€umen. Der Normbegriff ist noch sehr allgemein. Neben dem LĂ€ngenbegriff der euklidischen Geometrie enthĂ€lt er eine den stochastischen Matrizen angepaĂte Norm. Jede Norm auf VektorrĂ€umen induziert eine Norm fĂŒr lineare Abbildungen und Matrizen. So wird End(V) eine normierte Algebra. Die wichtigsten Ergebnisse im Abschnitt 6.2 sind der Ergodensatz 6.2.8 ĂŒber Kontraktionen und die Formel 6.2.10 fĂŒr den Spektralradius. Als Anwendung beweisen wir in 6.3 den Satz von Perron-Frobenius ĂŒber nichtnegative Matrizen, der Aussagen ĂŒber den Spektralradius und die zugehörenden Eigenvektoren macht. Ferner studieren wir in 6.4 die Exponentialfunktion von Matrizen und lösen Systeme von linearen Differentialgleichungen mit konstanten Koeffizienten. In 8.5 und 8.6 kommen wir darauf zurĂŒck und behandeln, dann ausgerĂŒstet mit der Eigenwerttheorie symmetrischer Matrizen, lineare Schwingungen. Mit Hilfe des Ergodensatzes bringen wir in 6.5 die Theorie der stochastischen Matrizen zu einem AbschluĂ und behandeln weitere Beispiele (Mischprozesse, Irrfahrten).
ZusammenfassungMit den HilbertrĂ€umen von endlicher Dimension ĂŒber â, den euklidischen VektorrĂ€umen, sind wir bei der klassischen Geometrie angekommen. Hier gibt es neben LĂ€ngen auch Winkel zwischen Vektoren. AusfĂŒhrlich behandeln wir âŠ
ZusammenfassungMit den HilbertrĂ€umen von endlicher Dimension ĂŒber â, den euklidischen VektorrĂ€umen, sind wir bei der klassischen Geometrie angekommen. Hier gibt es neben LĂ€ngen auch Winkel zwischen Vektoren. AusfĂŒhrlich behandeln wir die Isometrien euklidischer VektorrĂ€ume, die orthogonalen Abbildungen. Am Spezialfall der orthogonalen Gruppen schildern wir die Methode der infinitesimalen Abbildungen, die in der Lieschen Theorie eine zentrale Rolle spielt. Als Nebenprodukt erhalten wir einen natĂŒrlichen Zugang zum vektoriellen Produkt im â3. Wir fĂŒhren den Schiefkörper der Quaternionen ein und untersuchen mit seiner Hilfe die orthogonalen Gruppen in der Dimension drei und vier. Zum AbschluĂ bestimmen wir alle endlichen Untergruppen der orthogonalen Gruppe in der Dimension drei, wobei sich reizvolle ZusammenhĂ€nge mit den platonischen Körpern ergeben.
NĂŒtzliche Abbildungen auf Mengen mit algebraischen Strukturen sind solche, die die gegebenen Strukturen respektieren. FĂŒr die VektorrĂ€ume sind dies die linearen Abbildungen bzw. deren ĂŒbersetzung in die Sprache der Matrizen. âŠ
NĂŒtzliche Abbildungen auf Mengen mit algebraischen Strukturen sind solche, die die gegebenen Strukturen respektieren. FĂŒr die VektorrĂ€ume sind dies die linearen Abbildungen bzw. deren ĂŒbersetzung in die Sprache der Matrizen. Nach einer eingehenden Behandlung der Theorie gehen wir auf stochastische Matrizen als erste Anwendung ein. Diese spielen eine wichtige Rolle bei der Behandlung von sogenannten stochastischen Prozessen, etwa bei Mischprozessen, GlĂŒcksspielen und Modellen zur Genetik. Nach kurzen Abschnitten ĂŒber Spur, Projektionen und die zugehörigen Vektorraumzerlegungen folgt eine EinfĂŒhrung in die Codierungstheorie, die sich mit der Korrektur von zufĂ€lligen Fehlern bei der DatenĂŒbertragung beschĂ€ftigt. Neben den bis hierher entwickelten Grundtatsachen der Linearen Algebra spielen elementare AbzĂ€hlungen eine wichtige Rolle. Das Kapitel schlieĂt mit der Behandlung von elementaren Umformungen von Matrizen. Dies liefert Algorithmen zur Rangbestimmung und zum Lösen von linearen Gleichungssystemen, die uns immer wieder in den Anwendungen begegnen werden.
Der Vektorraum ist einer der zentralen Begriffe der Linearen Algebra. Da wir viele Anwendungen, insbesondere aus der Diskreten Mathematik, im Auge haben, betrachten wir nicht nur reelle VektorrĂ€ume, sondern solche âŠ
Der Vektorraum ist einer der zentralen Begriffe der Linearen Algebra. Da wir viele Anwendungen, insbesondere aus der Diskreten Mathematik, im Auge haben, betrachten wir nicht nur reelle VektorrĂ€ume, sondern solche ĂŒber beliebigen Körpern. Dies erfordert, daĂ wir nĂ€her auf algebraische Grundstrukturen eingehen. Wir beginnen mit dem Gruppenbegriff. In erster Linie fĂŒhren wir dabei eine Sprache ein, die spĂ€ter prĂ€gnante Formulierungen erlaubt. Es folgt der Ring- und Körperbegriff. Als Anwendung des Rings der ganz rationalen Zahlen modulo n besprechen wir das RSA-Verfahren aus der Kryptographie, das eine DatenĂŒbertragung gegen unerlaubten Zugriff seitens Dritter sichert. Den reellen Zahlkörper setzen wir im folgenden stets als bekannt voraus. Seine feineren Eigenschaften, die in der Analysis behandelt werden, spielen zunĂ€chst keine Rolle. Eingehend behandeln wir jedoch den komplexen Zahlkörper und beweisen einfachste Eigenschaften von endlichen Körpern.
In diesem Kapitel fĂŒhren wir Skalarprodukte auf VektorrĂ€umen ĂŒber beliebigen Körpern ein. Dies fĂŒhrt zu einem OrthogonalitĂ€tsbegriff und orthogonalen Zerlegungen. Auf die klassischen â- oder â-VektorrĂ€ume mit definitem Skalarprodukt gehen âŠ
In diesem Kapitel fĂŒhren wir Skalarprodukte auf VektorrĂ€umen ĂŒber beliebigen Körpern ein. Dies fĂŒhrt zu einem OrthogonalitĂ€tsbegriff und orthogonalen Zerlegungen. Auf die klassischen â- oder â-VektorrĂ€ume mit definitem Skalarprodukt gehen wir dann in den Kapiteln 8 und 9 ausfĂŒhrlich ein. Ab 7.3 interessieren uns VektorrĂ€ume mit isotropen Vektoren. Dazu geben wir zwei ganz verschiedene Anwendungen. In 7.4 verwenden wir fĂŒr endliche Körper K das kanonische Skalarprodukt auf $$ K^n $$ , um den Dualen eines Codes C †$$ K^n $$ zu definieren. Dies liefert weitere Beispiele von interessanten Codes und allgemeine Strukturaussagen. In 7.5 versehen wir den Vektorraum â4 mit einem indefiniten Skalarprodukt. Dies fĂŒhrt zum Minkowskiraum und seinen Isometrien, den Lorentz-Transformationen. Diese Ergebnisse wenden wir in 7.6 an, um die geometrischen Grundlagen der speziellen RelativitĂ€tstheorie von Einstein darzustellen. Die spezielle RelativitĂ€tstheorie von 1905 steht neben der Quantentheorie am Anfang der groĂen Revolutionen in der Physik des 20. Jahrhunderts, die die Vorstellungen von Raum und Zeit grundlegend verĂ€ndert haben.
The following conjecture is studied. Let $G$ be a simple nonabelian group. If $H$ is any group which has the same set of character degrees as $G$, then $H \cong âŠ
The following conjecture is studied. Let $G$ be a simple nonabelian group. If $H$ is any group which has the same set of character degrees as $G$, then $H \cong G \times A$, where $A$ is abelian. In the present paper this is proved if $G$ is a Suzuki group on some $SL(2,2^{f})$.
Notations and results from group theory representations and representation-modules simple and semisimple modules orthogonality relations the group algebra characters of abelian groups degrees of irreducible representations characters of some small âŠ
Notations and results from group theory representations and representation-modules simple and semisimple modules orthogonality relations the group algebra characters of abelian groups degrees of irreducible representations characters of some small groups products of representation and characters on the number of solutions gm =1 in a group a theorem of A. Hurwitz on multiplicative sums of squares permutation representations and characters the class number real characters and real representations Coprime action groups pa qb Fronebius groups induced characters Brauer's permutation lemma and Glauberman's character correspondence Clifford theory 1 projective representations Clifford theory 2 extension of characters Degree pattern and group structure monomial groups representation of wreath products characters of p-groups groups with a small number of character degrees linear groups the degree graph groups all of whose character degrees are primes two special degree problems lengths of conjugacy classes R. Brauer's theorem on the character ring applications of Brauer's theorems Artin's induction theorem splitting fields the Schur index integral representations three arithmetical applications small kernels and faithful irreducible characters TI-sets involutions groups whose Sylow-2-subgroups are generalized quaternion groups perfect Fronebius complements. (Part contents).
This is a report on research conducted at Mainz from 1984 to 1990, made possible by the DFG-project.
This is a report on research conducted at Mainz from 1984 to 1990, made possible by the DFG-project.
Algebras, modules, and representations Group representations and characters Characters and integrality Products of characters Induced characters Normal subgroups T.I. sets and exceptional characters Brauer's theorem Changing the field The Schur âŠ
Algebras, modules, and representations Group representations and characters Characters and integrality Products of characters Induced characters Normal subgroups T.I. sets and exceptional characters Brauer's theorem Changing the field The Schur index Projective representations Character degrees Character correspondence Linear groups Changing the characteristic Some character tables Bibliographic notes References Index.
In this paper we continue our study of the relationship between the structure of a finite group G and the set of degrees of its irreducible complex characters.The following hypotheses âŠ
In this paper we continue our study of the relationship between the structure of a finite group G and the set of degrees of its irreducible complex characters.The following hypotheses on the degrees are considered: (A) G has r.x. e for some prime p, i.e. all the degrees divide p e , (B) the degrees are linearly ordered by divisibility and all except 1 are divisible by exactly the same set of primes, (C) G has a.c.m, i.e., all the degrees except 1 are equal to some fixed m, (D) all the degrees except 1 are prime (not necessarily the same prime) and (E) all the degrees except 1 are divisible by p e > p but none is divisible by p e+1 .In each of these situations, group theoretic information is deduced from the character theoretic hypothesis and in several cases complete characterizations are obtained.
In 1896 G. Frobenius proved: the degree of any (absolutely) irreducible representation of a finite group divides its order. This theorem was improved by I. Schur in 1904 as follows: âŠ
In 1896 G. Frobenius proved: the degree of any (absolutely) irreducible representation of a finite group divides its order. This theorem was improved by I. Schur in 1904 as follows: the degree of any irreducible representation of a finite group divides the index of its centre.
The main result of this paper is the following: Theorem A. Let H and N be finite groups with coprime orders and suppose that H acts nontrivially on N via âŠ
The main result of this paper is the following: Theorem A. Let H and N be finite groups with coprime orders and suppose that H acts nontrivially on N via automorphisms. Assume that H fixes every nonlinear irreducible character of N. Then the derived subgroup of N is nilpotent and so N is solvable of nilpotent length ⊠2. Why might one be interested in a situation like this? There has been considerable interest in the question of what one can deduce about a group G from a knowledge of the set cd(G) = ïčx(l)lx â Irr(G) ïč of irreducible character degrees of G. Recently, attention has been focused on the prime divisors of the elements of cd(G). For instance, in [9], O. Manz and R. Staszewski consider Ï-separable groups (for some set Ï of primes) with the property that every element of cd(G) is either a 77-number or a Ï'-number.
The theory of group characters was originated by G. Frobenius and has been studied by many authors including, above all, I. Schur and W. Burnside. As to the modular theory âŠ
The theory of group characters was originated by G. Frobenius and has been studied by many authors including, above all, I. Schur and W. Burnside. As to the modular theory we owe it to recent works by R. Brauer and his collaborators, including T. Nakayama and C. Nesbitt, which clarified also the connections between the structure theory and the representation theory.
Let P be a p-group which acts faithfully as a group of automorphisms on solvable p'-group H.In this paper we discuss the existence of an element he H having a âŠ
Let P be a p-group which acts faithfully as a group of automorphisms on solvable p'-group H.In this paper we discuss the existence of an element he H having a "small" centralizer in P. We give two sufficient conditions for the strongest possible result, that is the existence of an element n centralized in P only by the identity.The first is that the pair (p,n(H)), where n(H) denotes the set of prime divisors of | H |, be nonexceptional or essentially that it involve no FermĂąt or Mersenne primes.The second is that the orbits in H under action of P all have size smaller than pp.In any case we show that for some element n, | (ÂŁP(/i) j g | P |1/2.We apply these results to two distinct subjects.First we generalize a theorem of Ito which concerns the minimum number of Sylow p-subgroups of a solvable group G which intersect to ÂŁ)p(G), the intersection of all of them.We show that if (p,7t(G)) is nonexceptional then there exists two such Sylow subgroups which intersect to DP(G).In any case there always exist three which work.Second, we obtain reduction theorems for the study of groups G all of whose absolutely irreducible representations have degrees which are powers of a prime p.In [3] certain relationships between the biggest of these degrees and the existence of "large" abelian subgroups of G were studied.We show here how in most cases these problems can be reduced to a study of p-groups.1. Nonexceptional prime pairs.Definition.Let p be a prime and n a set of primes.We say the pair (p, n) is nonexceptional provided it is not one of the following three types.(m,2): p is a Mersenne prime and 2en.(2,m): p = 2 and n contains a Mersenne prime.(2,/): p = 2 and n contains a FermĂąt prime.In this section we prove the following result.Theorem 1.1.Let p-group P act faithfully on solvable p'-group H. (i) If (p,n(H)) is nonexceptional then there exists he H with C/>(n) = {l}.(ii) If (p,n(H)) is not type (2,f) then there exists hj^^^eH with GP(n,) ndp(n2) = {1} and the product ÂĄÂŁP(hÂĄ)(iP(h2) being an abelian group.(iii) In any case there exists h1,h2eH with Gp(n,) oGP(n2) = {1}.
The object of this paper is to investigate the representation 21H induced by an irreducible representation 2t1 of a group G in an invariant subgroup H of G. In ?1 âŠ
The object of this paper is to investigate the representation 21H induced by an irreducible representation 2t1 of a group G in an invariant subgroup H of G. In ?1 it is shown that 21, is either itself irreducible or is fully reducible into conjugate irreducible representations of H. In ?2 it is shown that 2IG is imprimitive unless all the irreducible components of 21H are equivalent. In fact, if T is the representation space of Kg, and hence also of 21H , and if we lump together all equivalent subspaces of T under WH , then the resulting subspaces T1 , T2 , T. , * constitute a system of imprimitivity of WsG. If we define G' to be the subgroup of G leaving one of these invariant, say T, then the component of 21G' in T, is an irreducible representation 2<;, of G', and 21G is expressible very simply in terms of WG, . These results hold for any group G and any ground-field P. In ??3-5, however, we make the assumption that P is algebraically closed. In ?3 it is found that [' , is the direct product of two irreducible projective representations of G', one of which is actually a projective representation r of the factor-group G'/H. In ?4 some progress is made on the question of whether or not a given irreducible representation of H can be embedded in some irreducible representation of G, and in ?5 we consider all possible ways of doing this. Two irreducible representations 21G and TG of G are said to be associate if 21H and EH have an irreducible component in common; associates differ only in the projective representation r of G'/H mentioned above. In the case when the factor-group G/H is a finite cyclic group of order k, associates can be described as differing from each other only by a factor which is a one-dimensional representation of G/H, and hence just a kth root of unity. In the simplest case of all, when H is of index two in G, WG has just one associate 2* (besides itself) differing from 2tI only in that we change the sign of the matrices corresponding to elements of G not in H. The situation may then be described as follows. If 21G is not equivalent to * , then WH is irreducible. If t is equivalent to * , then 96f decomposes into two inequivalent (conjugate) irreducible components. If TG is another irreducible representation of G equivalent to neither Ads nor 2*, then TH can have no irreducible component in common with 2I(= 2H)Virtually all of this theory is known in the case of a finite group G, and the greater part of it goes back to Frobenius. For the decomposition of WH into conjugates we must refer to Frobenius' original paper.' For the results of ?2-
In 1903 H. F. Blichfeldt proved the following brilliant theorem : Let G be a matrix group of order g and of degree n. Let p be a prime divisor âŠ
In 1903 H. F. Blichfeldt proved the following brilliant theorem : Let G be a matrix group of order g and of degree n. Let p be a prime divisor of g such that Then G contains the abelian normal p -Sylow subgroup. In 1941 applying his modular theory of the group representation, R. Brauer improved this theorem in the case in which p divides g to the first power only. Further in 1943 H. F. Tuan improved this result of R. Brauer one step more.
Les Elements de mathematique de Nicolas Bourbaki ont pour objet une presentation rigoureuse, systematique et sans prerequis des mathematiques depuis leurs fondements. Ce premier volume du Livre sur les Groupes âŠ
Les Elements de mathematique de Nicolas Bourbaki ont pour objet une presentation rigoureuse, systematique et sans prerequis des mathematiques depuis leurs fondements. Ce premier volume du Livre sur les Groupes et algebre de Lie, neuvieme Livre du traite, est consacre aux concepts fondamentaux pour les algebres de Lie. Il comprend les paragraphes: - 1 Definition des algebres de Lie; 2 Algebre enveloppante d une algebre de Lie; 3 Representations; 4 Algebres de Lie nilpotentes; 5 Algebres de Lie resolubles; 6 Algebres de Lie semi-simples; 7 Le theoreme d Ado. Ce volume est une reimpression de l edition de 1971.
Let us consider soluble groups whose Sylow subgroups are all abelian. Such groups we call A -groups, following P. Hall. A -groups were investigated thoroughly by P. Hall and D. âŠ
Let us consider soluble groups whose Sylow subgroups are all abelian. Such groups we call A -groups, following P. Hall. A -groups were investigated thoroughly by P. Hall and D. R. Taunt from the view point of the structure theory. In this note, we want to give some remarks concerning representation theoretical properties of A -groups.
1. Dedekind Domains and Valuations.- 2. Algebraic Numbers and Integers.- 3. Units and Ideal Classes.- 4. Extensions.- 5. P-adic Fields.- 6. Applications of the Theory of P-adic Fields.- 7. Analytical âŠ
1. Dedekind Domains and Valuations.- 2. Algebraic Numbers and Integers.- 3. Units and Ideal Classes.- 4. Extensions.- 5. P-adic Fields.- 6. Applications of the Theory of P-adic Fields.- 7. Analytical Methods.- 8. Abelian Fields.- 9. Factorizations 9.1. 485Elementary Approach.- Appendix I. Locally Compact Abelian Groups.- Appendix II. Function Theory.- Appendix III. Baker's Method.- Problems.- References.- Author Index.- List of Symbols.
(1989). On the loewy series of the group algebra of groups of small p-Length. Communications in Algebra: Vol. 17, No. 5, pp. 1249-1274.
(1989). On the loewy series of the group algebra of groups of small p-Length. Communications in Algebra: Vol. 17, No. 5, pp. 1249-1274.
General results are provided on bounding the number of different prime factors of the order of finite groups in terms of the number for the order of elements.
General results are provided on bounding the number of different prime factors of the order of finite groups in terms of the number for the order of elements.