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{{Infobox scientist | |||
|name = Ferdinand Georg Frobenius | |||
|image = GeorgFrobenius.jpg | |||
|image_size = 150px | |||
|caption = Ferdinand Georg Frobenius | |||
|birth_date = {{birth date|df=y|1849|10|26}} | |||
|birth_place = [[Charlottenburg]] | |||
|death_date = {{death date and age|df=y|1917|08|3|1849|10|26}} | |||
|death_place = [[Berlin]] | |||
|nationality = [[Germans|German]] | |||
|field = [[Mathematics]] | |||
|work_institutions = [[Humboldt University of Berlin|University of Berlin]]<br>[[ETH Zurich]] | |||
|alma_mater = [[University of Göttingen]]<br>University of Berlin | |||
|doctoral_advisor = [[Karl Weierstrass]]<br>[[Ernst Kummer]] | |||
|doctoral_students = [[Richard Fuchs]]<br>[[Edmund Landau]]<br>[[Issai Schur]]<br>[[Konrad Knopp]]<br>[[Walter Schnee]] | |||
|known_for = [[Differential equations]]<br>[[Group theory]]<br>[[Cayley–Hamilton theorem]]<br>[[Frobenius method]] | |||
|influences = | |||
|influenced = | |||
|awards = | |||
}} | |||
'''Ferdinand Georg Frobenius''' (26 October 1849 – 3 August 1917) was a [[Germans|German]] [[mathematician]], best known for his contributions to the theory of [[elliptic functions]], [[differential equations]] and to [[group theory]]. He is known for the famous determinantal identities, known as Frobenius-Stickelberger formulae, governing elliptic functions, and for developing the theory of biquadratic forms. He was also the first to introduce the notion of rational approximations of functions (nowadays known as [[Padé approximants]]), and gave the first full proof for the [[Cayley–Hamilton theorem]]. He also lent his name to certain differential-geometric objects in modern mathematical physics, known as [[Frobenius manifolds]]. | |||
==Biography== | |||
Ferdinand Georg Frobenius was born on 26 October 1849 in [[Charlottenburg]], a suburb of [[Berlin]]<ref>{{cite web|url=http://www-history.mcs.st-and.ac.uk/BirthplaceMaps/Berlin.html|date=October 26, 2010|title=Born in Berlin}}</ref> from parents Christian Ferdinand Frobenius, a [[Protestant]] parson, and Christine Elizabeth Friedrich. He entered the Joachimsthal Gymnasium in 1860 when he was nearly eleven.<ref name="Bio">{{cite web|url=http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Frobenius.html|title=Biography|date=26 October 2010}}</ref> In 1867, after graduating, he went to the [[University of Göttingen]] where he began his university studies but he only studied there for one semester before returning to Berlin, where he attended lectures by [[Kronecker]], [[Ernst Eduard Kummer|Kummer]] and [[Karl Weierstrass]]. He received his doctorate (awarded with distinction) in 1870 supervised by Weierstrass. His thesis, supervised by [[Karl Weierstrass|Weierstrass]], was on the solution of differential equations. In 1874, after having taught at secondary school level first at the Joachimsthal Gymnasium then at the Sophienrealschule, he was appointed to the University of Berlin as an extraordinary professor of mathematics.<ref name="Bio" /> Frobenius was only in Berlin a year before he went to [[Zürich]] to take up an appointment as an ordinary professor at the [[ETH Zurich|Eidgenössische Polytechnikum]]. For seventeen years, between 1875 and 1892, Frobenius worked in Zürich. It was there that he married, brought up his family, and did much important work in widely differing areas of mathematics. In the last days of December 1891 Kronecker died and, therefore, his chair in Berlin became vacant. Weierstrass, strongly believing that Frobenius was the right person to keep Berlin in the forefront of mathematics, used his considerable influence to have Frobenius appointed. In 1893 he returned to Berlin, where he was elected to the [[Prussian Academy of Sciences]]. | |||
==Contributions to group theory== | |||
[[Group theory]] was one of Frobenius' principal interests in the second half of his career. One of his first contributions was the proof of the [[Sylow theorems]] for abstract groups. Earlier proofs had been for [[permutation group]]s. His proof of the first Sylow theorem (on the existence of Sylow groups) is one of those frequently used today. | |||
* Frobenius also has proved the following fundamental theorem: If a positive integer ''n'' divides the order |''G''| of a [[finite group]] ''G'', then the number of solutions of the equation ''x''<sup>''n''</sup> = 1 in ''G'' is equal to ''kn'' for some positive integer ''k''. He also posed the following problem: If, in the above theorem, ''k'' = 1, then the solutions of the equation ''x''<sup>''n''</sup> = 1 in ''G'' form a subgroup. Many years ago this problem was solved for [[solvable group]]s.<ref>Marshall Hall, Jr., ''The Theory of Groups'', 2nd ed. (Providence, Rhode Island : AMS Chelsea Publishing, 1999), pages 145-146, [http://books.google.com/books?id=oyxnWF9ssI8C&pg=PA145#v=onepage&q&f=false Theorem 9.4.1.]</ref> Only in 1991, after the [[classification of finite simple groups]], was this problem solved in general. | |||
More important was his creation of the theory of [[Character theory|group characters]] and [[group representation]]s, which are fundamental tools for studying the structure of groups. This work led to the notion of [[Character theory|Frobenius reciprocity]] and the definition of what are now called [[Frobenius group]]s. A group ''G'' is said to be a Frobenius group if there is a subgroup ''H'' < ''G'' such that | |||
:<math>H\cap H^x=\{1\}</math> for all <math>x\in G-H</math>. | |||
In that case, the set | |||
:<math>N=G-\bigcup_{x\in G-H}H^x</math> | |||
together with the identity element of ''G'' forms a subgroup which is [[nilpotent group|nilpotent]] as Thompson showed in his PhD thesis. All known proofs of that theorem make use of characters. In his first paper about characters (1896), Frobenius constructed the character table of the group <math>PSL(2,p)</math> of order (1/2)(''p''<sup>3</sup> − p) for all odd primes ''p'' (this group is simple provided ''p'' > 3). He also | |||
made fundamental contributions to the [[representation theory of the symmetric and alternating groups]]. | |||
==Contributions to number theory== | |||
Frobenius introduced a canonical way of turning primes into [[conjugacy classes]] in [[Galois group]]s over '''Q'''. Specifically, if ''K''/'''Q''' is a finite Galois extension then to each (positive) prime ''p'' which does not [[ramification|ramify]] in ''K'' and to each prime ideal ''P'' lying over ''p'' in ''K'' there is a unique element ''g'' of Gal(''K''/'''Q''') satisfying the condition ''g''(''x'') = ''x''<sup>''p''</sup> (mod ''P'') for all integers ''x'' of ''K''. Varying ''P'' over ''p'' changes ''g'' into a conjugate (and every conjugate of ''g'' occurs in this way), so the conjugacy class of ''g'' in the Galois group is canonically associated to ''p''. This is called the Frobenius conjugacy class of ''p'' and any element of | |||
the conjugacy class is called a Frobenius element of ''p''. If we take for ''K'' the ''m''th [[cyclotomic field]], whose Galois group over '''Q''' is the units modulo ''m'' (and thus | |||
is abelian, so conjugacy classes become elements), then for ''p'' not dividing ''m'' the Frobenius class in the Galois group is ''p'' mod ''m''. From this point of view, | |||
the distribution of Frobenius conjugacy classes in Galois groups over '''Q''' (or, more generally, Galois groups over any number field) generalizes Dirichlet's classical result about primes in arithmetic progressions. The study of Galois groups of infinite-degree extensions of '''Q''' depends crucially on this construction of Frobenius elements, which provides in a sense a dense subset of elements which are accessible to detailed study. | |||
==See also== | |||
*[[List of things named after Ferdinand Georg Frobenius]] | |||
==Publications== | |||
*{{Citation | last1=Frobenius | first1=Ferdinand Georg | author1-link=Ferdinand Georg Frobenius | title=Gesammelte Abhandlungen. Bände I, II, III | publisher=[[Springer-Verlag]] | location=Berlin, New York | editor-first= J.-P.|editor-last= Serre | isbn=978-3-540-04120-7 | mr=0235974 | year=1968}} | |||
==References== | |||
{{Reflist}} | |||
*{{Citation | last1=Curtis | first1=Charles W. | authorlink = Charles W. Curtis | title=Pioneers of Representation Theory: Frobenius, Burnside, Schur, and Brauer | url=http://books.google.com/books?isbn=0821826778 | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=History of Mathematics | isbn=978-0-8218-2677-5 | mr=1715145 | year=2003}} [http://www.ams.org/journals/bull/2000-37-03/S0273-0979-00-00867-3/ Review] | |||
==External links== | |||
*{{MacTutor Biography|id=Frobenius}} | |||
* {{MathGenealogy|id=4642}} | |||
*[http://neo-classical-physics.info/uploads/3/0/6/5/3065888/frobenius_-_hypercomplex_i.pdf G. Frobenius, "Theory of hypercomplex quantities"] (English translation) | |||
{{Authority control|PND=119045605|LCCN=n/84/801120|VIAF=56675988}} | |||
{{Persondata <!-- Metadata: see [[Wikipedia:Persondata]]. --> | |||
| NAME = Frobenius, Ferdinand Georg | |||
| ALTERNATIVE NAMES = | |||
| SHORT DESCRIPTION = German mathematician | |||
| DATE OF BIRTH = 26 October 1849 | |||
| PLACE OF BIRTH = [[Charlottenburg]] | |||
| DATE OF DEATH = 31 August 1917 | |||
| PLACE OF DEATH = [[Berlin]] | |||
}} | |||
{{DEFAULTSORT:Frobenius, Ferdinand Georg}} | |||
[[Category:1849 births]] | |||
[[Category:1917 deaths]] | |||
[[Category:19th-century German mathematicians]] | |||
[[Category:20th-century mathematicians]] | |||
[[Category:German mathematicians]] | |||
[[Category:Group theorists]] | |||
[[Category:Members of the Prussian Academy of Sciences]] | |||
[[Category:People from Berlin]] | |||
[[Category:People from the Province of Brandenburg]] | |||
[[Category:University of Göttingen alumni]] | |||
[[Category:Humboldt University of Berlin alumni]] | |||
[[Category:Humboldt University of Berlin faculty]] | |||
[[Category:ETH Zurich faculty]] | |||
Revision as of 01:54, 29 January 2014
Template:Infobox scientist Ferdinand Georg Frobenius (26 October 1849 – 3 August 1917) was a German mathematician, best known for his contributions to the theory of elliptic functions, differential equations and to group theory. He is known for the famous determinantal identities, known as Frobenius-Stickelberger formulae, governing elliptic functions, and for developing the theory of biquadratic forms. He was also the first to introduce the notion of rational approximations of functions (nowadays known as Padé approximants), and gave the first full proof for the Cayley–Hamilton theorem. He also lent his name to certain differential-geometric objects in modern mathematical physics, known as Frobenius manifolds.
Biography
Ferdinand Georg Frobenius was born on 26 October 1849 in Charlottenburg, a suburb of Berlin[1] from parents Christian Ferdinand Frobenius, a Protestant parson, and Christine Elizabeth Friedrich. He entered the Joachimsthal Gymnasium in 1860 when he was nearly eleven.[2] In 1867, after graduating, he went to the University of Göttingen where he began his university studies but he only studied there for one semester before returning to Berlin, where he attended lectures by Kronecker, Kummer and Karl Weierstrass. He received his doctorate (awarded with distinction) in 1870 supervised by Weierstrass. His thesis, supervised by Weierstrass, was on the solution of differential equations. In 1874, after having taught at secondary school level first at the Joachimsthal Gymnasium then at the Sophienrealschule, he was appointed to the University of Berlin as an extraordinary professor of mathematics.[2] Frobenius was only in Berlin a year before he went to Zürich to take up an appointment as an ordinary professor at the Eidgenössische Polytechnikum. For seventeen years, between 1875 and 1892, Frobenius worked in Zürich. It was there that he married, brought up his family, and did much important work in widely differing areas of mathematics. In the last days of December 1891 Kronecker died and, therefore, his chair in Berlin became vacant. Weierstrass, strongly believing that Frobenius was the right person to keep Berlin in the forefront of mathematics, used his considerable influence to have Frobenius appointed. In 1893 he returned to Berlin, where he was elected to the Prussian Academy of Sciences.
Contributions to group theory
Group theory was one of Frobenius' principal interests in the second half of his career. One of his first contributions was the proof of the Sylow theorems for abstract groups. Earlier proofs had been for permutation groups. His proof of the first Sylow theorem (on the existence of Sylow groups) is one of those frequently used today.
- Frobenius also has proved the following fundamental theorem: If a positive integer n divides the order |G| of a finite group G, then the number of solutions of the equation xn = 1 in G is equal to kn for some positive integer k. He also posed the following problem: If, in the above theorem, k = 1, then the solutions of the equation xn = 1 in G form a subgroup. Many years ago this problem was solved for solvable groups.[3] Only in 1991, after the classification of finite simple groups, was this problem solved in general.
More important was his creation of the theory of group characters and group representations, which are fundamental tools for studying the structure of groups. This work led to the notion of Frobenius reciprocity and the definition of what are now called Frobenius groups. A group G is said to be a Frobenius group if there is a subgroup H < G such that
In that case, the set
together with the identity element of G forms a subgroup which is nilpotent as Thompson showed in his PhD thesis. All known proofs of that theorem make use of characters. In his first paper about characters (1896), Frobenius constructed the character table of the group of order (1/2)(p3 − p) for all odd primes p (this group is simple provided p > 3). He also made fundamental contributions to the representation theory of the symmetric and alternating groups.
Contributions to number theory
Frobenius introduced a canonical way of turning primes into conjugacy classes in Galois groups over Q. Specifically, if K/Q is a finite Galois extension then to each (positive) prime p which does not ramify in K and to each prime ideal P lying over p in K there is a unique element g of Gal(K/Q) satisfying the condition g(x) = xp (mod P) for all integers x of K. Varying P over p changes g into a conjugate (and every conjugate of g occurs in this way), so the conjugacy class of g in the Galois group is canonically associated to p. This is called the Frobenius conjugacy class of p and any element of the conjugacy class is called a Frobenius element of p. If we take for K the mth cyclotomic field, whose Galois group over Q is the units modulo m (and thus is abelian, so conjugacy classes become elements), then for p not dividing m the Frobenius class in the Galois group is p mod m. From this point of view, the distribution of Frobenius conjugacy classes in Galois groups over Q (or, more generally, Galois groups over any number field) generalizes Dirichlet's classical result about primes in arithmetic progressions. The study of Galois groups of infinite-degree extensions of Q depends crucially on this construction of Frobenius elements, which provides in a sense a dense subset of elements which are accessible to detailed study.
See also
Publications
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References
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Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
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Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
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15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
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External links
- Template:MacTutor Biography
- 50 yr old Print Journalist Broadbent from Deep River, has hobbies and interests which includes r/c helicopters, property developers in singapore house for rent and scrabble. Suggests that you go to Tomb of Askia.
- G. Frobenius, "Theory of hypercomplex quantities" (English translation)
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- ↑ Template:Cite web
- ↑ 2.0 2.1 Template:Cite web
- ↑ Marshall Hall, Jr., The Theory of Groups, 2nd ed. (Providence, Rhode Island : AMS Chelsea Publishing, 1999), pages 145-146, Theorem 9.4.1.
- 1849 births
- 1917 deaths
- 19th-century German mathematicians
- 20th-century mathematicians
- German mathematicians
- Group theorists
- Members of the Prussian Academy of Sciences
- People from Berlin
- People from the Province of Brandenburg
- University of Göttingen alumni
- Humboldt University of Berlin alumni
- Humboldt University of Berlin faculty
- ETH Zurich faculty