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Otto Grün
Otto Grün was a German mathematician. He was known primarily for his work in the theory of finite groups, which have often been praised for being foundational to further developments in the field. He was born on June 26, 1888, in Berlin. He obtained his Ph.D. from Humboldt University of Berlin in 1948 under the advisory of Hermann Ludwig Schmid. He died in October 1974 on or before the 10th. See also *Grün's lemma In mathematics, more specifically in group theory, a group is said to be perfect if it equals its own commutator subgroup, or equivalently, if the group has no non-trivial abelian quotients. Examples The smallest (non-trivial) perfect group is ... * Grün's first theorem References Bibliography * 20th-century German mathematicians 1888 births 1974 deaths Humboldt University of Berlin alumni {{Mathematician-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Berlin
Berlin ( ; ) is the Capital of Germany, capital and largest city of Germany, by both area and List of cities in Germany by population, population. With 3.7 million inhabitants, it has the List of cities in the European Union by population within city limits, highest population within its city limits of any city in the European Union. The city is also one of the states of Germany, being the List of German states by area, third smallest state in the country by area. Berlin is surrounded by the state of Brandenburg, and Brandenburg's capital Potsdam is nearby. The urban area of Berlin has a population of over 4.6 million and is therefore the most populous urban area in Germany. The Berlin/Brandenburg Metropolitan Region, Berlin-Brandenburg capital region has around 6.2 million inhabitants and is Germany's second-largest metropolitan region after the Rhine-Ruhr region, as well as the List of EU metropolitan areas by GDP, fifth-biggest metropolitan region by GDP in the European Union. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Germany
Germany, officially the Federal Republic of Germany, is a country in Central Europe. It lies between the Baltic Sea and the North Sea to the north and the Alps to the south. Its sixteen States of Germany, constituent states have a total population of over 84 million in an area of , making it the most populous member state of the European Union. It borders Denmark to the north, Poland and the Czech Republic to the east, Austria and Switzerland to the south, and France, Luxembourg, Belgium, and the Netherlands to the west. The Capital of Germany, nation's capital and List of cities in Germany by population, most populous city is Berlin and its main financial centre is Frankfurt; the largest urban area is the Ruhr. Settlement in the territory of modern Germany began in the Lower Paleolithic, with various tribes inhabiting it from the Neolithic onward, chiefly the Celts. Various Germanic peoples, Germanic tribes have inhabited the northern parts of modern Germany since classical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Humboldt University Of Berlin
The Humboldt University of Berlin (, abbreviated HU Berlin) is a public research university in the central borough of Mitte in Berlin, Germany. The university was established by Frederick William III on the initiative of Wilhelm von Humboldt, Johann Gottlieb Fichte and Friedrich Daniel Ernst Schleiermacher as the University of Berlin () in 1809, and opened in 1810. From 1828 until its closure in 1945, it was named the (Royal) Friedrich Wilhelm University of Berlin (FWU Berlin; ). During the Cold War, the university found itself in East Berlin and was ''de facto'' split in two when the Free University of Berlin opened in West Berlin. The university received its current name in honour of Alexander and Wilhelm von Humboldt in 1949. The university is divided into nine faculties including its medical school shared with the Freie Universität Berlin. The university has a student enrollment of around 35,000 students, and offers degree programs in some 171 disciplines from un ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematical model, models, and mathematics#Calculus and analysis, change. History One of the earliest known mathematicians was Thales of Miletus (); he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed. He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales's theorem. The number of known mathematicians grew when Pythagoras of Samos () established the Pythagorean school, whose doctrine it was that mathematics ruled the universe and whose motto was "All is number". It was the Pythagoreans who coined the term "mathematics", and with whom the study of mathematics for its own sake begins. The first woman math ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Groups
In abstract algebra, a finite group is a group (mathematics), group whose underlying set is finite set, finite. Finite groups often arise when considering symmetry of Symmetry in mathematics, mathematical or Symmetry (physics), physical objects, when those objects admit just a finite number of structure-preserving transformations. Important examples of finite groups include cyclic groups and permutation groups. The study of finite groups has been an integral part of group theory since it arose in the 19th century. One major area of study has been classification: the classification of finite simple groups (those with no nontrivial normal subgroup) was completed in 2004. History During the twentieth century, mathematicians investigated some aspects of the theory of finite groups in great depth, especially the Local analysis, local theory of finite groups and the theory of Solvable group, solvable and nilpotent groups. As a consequence, the complete classification of finite sim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perfect Group
In mathematics, more specifically in group theory, a group is said to be perfect if it equals its own commutator subgroup, or equivalently, if the group has no non-trivial abelian quotients. Examples The smallest (non-trivial) perfect group is the alternating group ''A''5. More generally, any non-abelian simple group is perfect since the commutator subgroup is a normal subgroup with abelian quotient. However, a perfect group need not be simple; for example, the special linear group over the field with 5 elements, SL(2,5) (or the binary icosahedral group, which is isomorphic to it) is perfect but not simple (it has a non-trivial center containing -\!\left(\begin1 & 0 \\ 0 & 1\end\right) = \left(\begin4 & 0 \\ 0 & 4\end\right)). The direct product of any two simple non-abelian groups is perfect but not simple; the commutator of two elements is ''a'',''b''),(''c'',''d'')= ( 'a'',''c'' 'b'',''d''. Since commutators in each simple group form a generating set, pairs of com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Focal Subgroup Theorem
In abstract algebra, the focal subgroup theorem describes the fusion of elements in a Sylow subgroup of a finite group. The focal subgroup theorem was introduced in and is the "first major application of the transfer" according to . The focal subgroup theorem relates the ideas of transfer and fusion such as described by Otto Grün in . Various applications of these ideas include local criteria for p-nilpotent group, ''p''-nilpotence and various non-simple group, simplicity criteria focussing on showing that a finite group has a normal subgroup of Index of a subgroup, index ''p''. Background The focal subgroup theorem relates several lines of investigation in finite group theory: normal subgroups of index a power of ''p'', the transfer homomorphism, and fusion of elements. Subgroups The following three normal subgroups of index a power of ''p'' are naturally defined, and arise as the smallest normal subgroups such that the quotient is (a certain kind of) ''p''-group. Formal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1888 Births
Events January * January 3 – The great telescope (with an objective lens of diameter) at Lick Observatory in California is first used. * January 12 – The Schoolhouse Blizzard hits Dakota Territory and the states of Montana, Minnesota, Nebraska, Kansas and Texas, leaving 235 dead, many of them children on their way home from school. * January 13 – The National Geographic Society is founded in Washington, D.C. * January 19 – The Battle of the Grapevine Creek, the last major conflict of the Hatfield–McCoy feud in the Southeastern United States. * January 21 – The Amateur Athletic Union is founded by William Buckingham Curtis in the United States. * January 26 – The Lawn Tennis Association is founded in England. February * February 27 – In West Orange, New Jersey, Thomas Edison meets with Eadweard Muybridge, who proposes a scheme for sound film. March * March 8 – The Agriculture College of Utah (later Utah State University) i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1974 Deaths
Major events in 1974 include the aftermath of the 1973 oil crisis and the resignation of President of the United States, United States President Richard Nixon following the Watergate scandal. In the Middle East, the aftermath of the 1973 Yom Kippur War determined politics; following List of Prime Ministers of Israel, Israeli Prime Minister Golda Meir's resignation in response to high Israeli casualties, she was succeeded by Yitzhak Rabin. In Europe, the Turkish invasion of Cyprus, invasion and occupation of northern Cyprus by Turkey, Turkish troops initiated the Cyprus dispute, the Carnation Revolution took place in Portugal, the Greek junta's collapse paves the way for the establishment of a Metapolitefsi, parliamentary republic and Chancellor of Germany, Chancellor of West Germany Willy Brandt resigned following an Guillaume affair, espionage scandal surrounding his secretary Günter Guillaume. In sports, the year was primarily dominated by the 1974 FIFA World Cup, FIFA World ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
