Mecklenburg-Vorpommern known by its anglicized name Mecklenburg-Western Pomerania, is a state of Germany. Of the country's 16 states, Mecklenburg-Vorpommern ranks 14th in population, 6th in area, 16th in population density. Schwerin is the state capital and Rostock is the largest city. Other major cities include Neubrandenburg, Greifswald, Wismar and Güstrow; the state of Mecklenburg-Vorpommern was established in 1945 after World War II through the merger of the historic regions of Mecklenburg and the Prussian Western Pomerania by the Soviet military administration in Allied-occupied Germany. Mecklenburg-Vorpommern became part of the German Democratic Republic in 1947, but was dissolved in 1952 during administrative reforms and its territory divided into the districts of Rostock and Neubrandenburg. Mecklenburg-Vorpommern was re-established in 1990 following German reunification, became one of the Federal Republic of Germany's new states. Mecklenburg-Vorpommern's coastline on the Baltic Sea features many holiday resorts and much unspoilt nature, including the islands such as Rügen and Usedom, as well as the Mecklenburg Lake District, making the state one of Germany's leading tourist destinations.

Three of Germany's fourteen national parks, as well as several hundred nature conservation areas, are in Mecklenburg-Vorpommern. The University of Rostock, established in 1419, the University of Greifswald, established in 1456, are among the oldest universities in Europe. Mecklenburg-Vorpommern was the site of the 33rd G8 summit in 2007. Due to its lengthy name, the state is abbreviated as MV or shortened to MeckPomm. In English, it is sometimes translated as "Mecklenburg-Western Pomerania". Inhabitants are called either Mecklenburger or Pomeranians, the combined form is never used; the full name in German is pronounced. Sometimes, Mecklenburg is pronounced; this is. Mecklenburg however is within the historical Low German language area, the "c" appeared in its name during the period of transition to Standard, High German usage; the introduction of the "c" is explained as follows: Either the "c" signals the stretched pronunciation of the preceding "e", or it signals the pronunciation of the subsequent "k" as an occlusive to prevent it from falsely being rendered as a fricative following a Low German trend.

Another explanation is that the "c" comes from a mannerism in High German officialese of writing unnecessary letters, a so-called Letternhäufelung. In the aftermath of the Second World War and German reunification in 1990, the state was constituted from the historic region of Mecklenburg and Western Pomerania, both of which had long and rich independent histories. Human settlement in the area of modern Mecklenburg and Vorpommern began after the Ice Age, about 10,000 BC. About two thousand years ago, Germanic peoples were recorded in the area. Most of them left during the Migration Period, heading towards Spain and France, leaving the area deserted. In the 6th century Polabian Slavs populated the area. While Mecklenburg was settled by the Obotrites, Vorpommern was settled by the Rani. Along the coast and Slavs established trade posts like Reric and Menzlin. In the 12th century and Vorpommern were conquered by Henry the Lion and incorporated into the Duchy of Saxony, joining the Holy Roman Empire in the 1180s.

Parts of Mecklenburg-Vorpommern was settled with Germans in the Ostsiedlung process, starting in the 12th century. In the late 12th century, Henry the Lion, Duke of the Saxons, conquered the Obotrites, subjugated its Nikloting dynasty, Christianized its people. In the course of time, German monks, nobility and traders arrived to settle here. After the 12th century, the territory remained stable and independent of its neighbours. Mecklenburg first became a duchy of the Holy Roman Empire in 1348. Though partitioned and re-partitioned within the same dynasty, Mecklenburg always shared a common history and identity; the states of Mecklenburg-Schwerin and Mecklenburg-Strelitz became Grand Duchies in 1815, in 1870 they voluntarily joined the new German Empire, while retaining their own internal autonomy. After the First World War and the abdication of the German Kaiser, the monarchies of the duchies were abolished and republican governments of both Mecklenburg states were established, until the Nazi government merged the two states into a unified state of Mecklenburg, a meaningless administrative decision under the centralised regime.

Vorpommern Fore-Pomerania, is the smaller, western part of the former Prussian Province of Pomerania. In the Middle Ages, the area was ruled by the Pomeranian dukes as part of the Duchy of Pomerania. Pomerania was under Swedish rule after the Peace of Westphalia from 1648 until 1815 as Swedish Pomerania. Pomerania became a province of Prussia in 1815 and remained so until 1945. In May 1945, the armies of the Soviet Union and the Western allies met east of Schwerin. Following the Potsdam Agreement, the Western allies handed over Mecklenburg to the Soviets. Mecklenburg-West Pomerania was established on 9 July 1945, by order No. 5 of Red Army Marshal Georgy Zhukov, head of the Soviet Military Admini

Ján Popluhár

Ján Popluhár was a former Slovak football player who played for club ŠK Slovan Bratislava and was a member of the Czechoslovakia national team, playing in two World Cups. Popluhár started his football career after high school with RH Brno. However, he is best known for the 15 seasons he spent with ŠK Slovan Bratislava where he made 262 competitive appearances in defence, scoring 21 goals, following further stints with Brno and Zbrojovka Brno finished on 306 Czechoslovak league games and a goalscoring tally of 24. After finishing his career with Slovan in 1968 he moved to French league side Olympique Lyonnais. Two seasons there were enough, but subsequently he spent five years with Austrian amateur club SK Slovan Vienna as player/coach. Many of the thousands of fans in Vienna thought that Popluhár would be shown up by players twenty years younger than he was, but this did not happen. However, at the age of 44 Popluhár realised. Slovan Bratislava coach Leopold Šťastný was famous for creating nicknames for his players, so Popluhár became known in Slovak football circles as'Bimbo'.

"I was called this because I always looked, indeed was, good-natured, I would never hurt anybody", recalls Popluhár. At the 1962 World Cup in Chile, instead of turning the situation to his team's advantage, Popluhár brought the referee's attention to the injured Pelé. In 1997, he was awarded the World Fair Play award for this sporting act. "I met Pelé several times, the first occasion in Chile and in various international and club games. He was undoubtedly the best, but there were not many one-to-one situations he won and not many Slovak players who scored in the famous Maracana stadium. I belong to this lucky group after I scored with a free kick against Brazil in June 1966", he remembers. Czechoslovak football was well represented in 1963 at Wembley Stadium in a match to mark the centenary of the English Football Association. Popluhár, along with Svatopluk Pluskal and Josef Masopust, played in a world team that included Alfredo Di Stéfano, Raymond Kopa, Uwe Seeler, Denis Law, Eusébio and Ferenc Puskás against an England eleven.

Further appearances in all-star teams followed. Popluhár was elected Footballer of the Year in the former Czechoslovakia for the first time in 1965, despite the great form of his Czechoslovak teammate Josef Masopust. "When I was at my peak, the conditions for a football player were modest here in Czechoslovakia, but the spirit and support of the spectators was incredible. I am not sure that today's players, would want to return to my era", he said. After working for a sports company that went bankrupt and suffering prolonged health problems resulting in a modest invalidity pension and an unsuitable job, he returned to his footballing roots. World Soccer World XI: 1964, 1967, 1968 Czechoslovak Footballer of the Year: 1965 In 2000 Popluhár was elected as the best Slovak footballer of the 20th century. UEFA Slovak Golden Player: 2003 Asteroid 267585 Popluhár, discovered by NEAT at Palomar Observatory in 2002, was named in his honor based on a suggestion by Slovak amateur astronomer Stefan Kürti.

The official naming citation was published by the Minor Planet Center on 15 June 2011. Czechoslovak Cup: 1962, 1963, 1968 1958 FIFA World Cup: Group stage 1960 European Nations' Cup: Bronze Medal 1962 FIFA World Cup: Silver Medal Ján Popluhár at FAČR – Slovakia's Golden Player – Popluhár: The people's choice Ján Popluhár at

Generalized dihedral group

In mathematics, the generalized dihedral groups are a family of groups with algebraic structures similar to that of the dihedral groups. They include the finite dihedral groups, the infinite dihedral group, the orthogonal group O. For any abelian group H, the generalized dihedral group of H, written Dih, is the semidirect product of H and Z2, with Z2 acting on H by inverting elements. I.e. D i h = H ⋊ ϕ Z 2 with φ the identity and φ inversion, thus we get: * = * = for all h1, h2 in H and t2 in Z2. Note that * =, i.e. first the inversion and the operation in H. * =. The subgroup of Dih of elements is a normal subgroup of index 2, isomorphic to H, while the elements are all their own inverse; the conjugacy classes are: the sets the sets Thus for every subgroup M of H, the corresponding set of elements is a normal subgroup. We have: Dih / M = Dih Dihn = Dih For n there are two sets, each generates a normal subgroup of type Dihn / 2; as subgroups of the isometry group of the set of vertices of a regular n-gon they are different: the reflections in one subgroup all have two fixed points, while none in the other subgroup has.

However, they are isomorphic as abstract groups. For odd n there is only one set Dih∞ = Dih; as subgroups of the isometry group of Z they are different: the reflections in one subgroup all have a fixed point, the mirrors are at the integers, while none in the other subgroup has, the mirrors are in between. However, they are isomorphic as abstract groups. Dih, or orthogonal group O, or O: the isometry group of a circle, or equivalently, the group of isometries in 2D that keep the origin fixed; the rotations form the circle group S1, or equivalently SO written SO, R/Z. In the latter case one of the reflections is complex conjugation. There are no proper normal subgroups with reflections; the discrete normal subgroups are cyclic groups of order n for all positive integers n. The quotient groups are isomorphic with the same group Dih. Dih: the group of isometries of Rn consisting of all translations and inversion in all points. H can be any subgroup of Rn. Discrete subgroups of Dih which contain translations in one direction are of frieze group type ∞ ∞ and 22 ∞.

Discrete subgroups of Dih which contain translations in two directions are of wallpaper group type p1 and p2. Discrete subgroups of Dih which contain translations in three directions are space groups of the triclinic crystal system. Dih is Abelian, with the semidirect product a direct product, if and only if all elements of H are their own inverse, i.e. an elementary abelian 2-group: Dih = Dih1 = Z2 Dih = Dih2 = Z2 × Z2 Dih = Dih2 × Z2 = Z2 × Z2 × Z2etc. Dih and its dihedral subgroups are disconnected topological groups. Dih consists of two connected components: the identity component isomorphic to Rn, the component with the reflections. O consists of two connected components: the identity component isomorphic to the circle group, the component with the reflections. For the group Dih∞ we can distinguish two cases: Dih∞ as the isometry group of Z Dih∞ as a 2-dimensional isometry group generated by a rotation by an irrational number of turns, a reflectionBoth topological groups are disconnected, but in the first case the components are open, while in the second case they are not.

The first topological group is a closed subgroup of Dih but the second is not a closed subgroup of O