derek-thomas/squad-v1.1-t5-question-generation huggingface dataset:
Matrix groups consist of matrices together with matrix multiplication. The general linear group GL(n, R) consists of all invertible n-by-n matrices with real entries. Its subgroups are referred to as matrix groups or linear groups. The dihedral group example mentioned above can be viewed as a (very small) matrix group. Another important matrix group is the special orthogonal group SO(n). It describes all possible rotations in n dimensions. Via Euler angles, rotation matrices are used in computer graphics.

Question: What groups combine matrices with matrix multiplication? Question: What includes all invertible n-by-n matrices with real entries? Question: What term describes subgroups of the general linear group? Question: What matrix group portrays all possible rotations in n dimensions? Question: Rotation matrix groups are utilized in computer graphics with what concept?
Exchanging "+" and "−" in the expression, i.e. permuting the two solutions of the equation can be viewed as a (very simple) group operation. Similar formulae are known for cubic and quartic equations, but do not exist in general for degree 5 and higher. Abstract properties of Galois groups associated with polynomials (in particular their solvability) give a criterion for polynomials that have all their solutions expressible by radicals, i.e. solutions expressible using solely addition, multiplication, and roots similar to the formula above.

Question: What degree does not include simple formulas for cubic and quatric equations? Question: What concept is associated with the solvability of polynomials? Question: What are used to express the solutions of polynomials?
A group is called finite if it has a finite number of elements. The number of elements is called the order of the group. An important class is the symmetric groups SN, the groups of permutations of N letters. For example, the symmetric group on 3 letters S3 is the group consisting of all possible orderings of the three letters ABC, i.e. contains the elements ABC, ACB, ..., up to CBA, in total 6 (or 3 factorial) elements. This class is fundamental insofar as any finite group can be expressed as a subgroup of a symmetric group SN for a suitable integer N (Cayley's theorem). Parallel to the group of symmetries of the square above, S3 can also be interpreted as the group of symmetries of an equilateral triangle.

Question: What does a finite group include? Question: What is the number of elements in a group named? Question: What type of class has a finite group that can be expressed as a subgroup of a symmetric group? Question: What can be described as the group of symmetries of an equilateral triangle?
Mathematicians often strive for a complete classification (or list) of a mathematical notion. In the context of finite groups, this aim leads to difficult mathematics. According to Lagrange's theorem, finite groups of order p, a prime number, are necessarily cyclic (abelian) groups Zp. Groups of order p2 can also be shown to be abelian, a statement which does not generalize to order p3, as the non-abelian group D4 of order 8 = 23 above shows. Computer algebra systems can be used to list small groups, but there is no classification of all finite groups.q[›] An intermediate step is the classification of finite simple groups.r[›] A nontrivial group is called simple if its only normal subgroups are the trivial group and the group itself.s[›] The Jordan–Hölder theorem exhibits finite simple groups as the building blocks for all finite groups. Listing all finite simple groups was a major achievement in contemporary group theory. 1998 Fields Medal winner Richard Borcherds succeeded in proving the monstrous moonshine conjectures, a surprising and deep relation between the largest finite simple sporadic group—the "monster group"—and certain modular functions, a piece of classical complex analysis, and string theory, a theory supposed to unify the description of many physical phenomena.

Question: What level of finality do mathematicians try to reach with math concepts? Question: What area of classification leads to troublesome arithmetic? Question: What defines finite groups of order p, a prime number, as being necessarily cyclic (abelian) groups Zp? Question: What can be used to classify small groups even though there is no classification of all finite groups? Question: What describes finite simple groups as the building pieces for all finite groups?
Some topological spaces may be endowed with a group law. In order for the group law and the topology to interweave well, the group operations must be continuous functions, that is, g • h, and g−1 must not vary wildly if g and h vary only little. Such groups are called topological groups, and they are the group objects in the category of topological spaces. The most basic examples are the reals R under addition, (R ∖ {0}, ·), and similarly with any other topological field such as the complex numbers or p-adic numbers. All of these groups are locally compact, so they have Haar measures and can be studied via harmonic analysis. The former offer an abstract formalism of invariant integrals. Invariance means, in the case of real numbers for example:

Question: What group operations must occur for group law and topology to integrate well? Question: What is the most simple example of topological groups? Question: What variables do locally compact groups share that can be studied by harmonic analysis?
for any constant c. Matrix groups over these fields fall under this regime, as do adele rings and adelic algebraic groups, which are basic to number theory. Galois groups of infinite field extensions such as the absolute Galois group can also be equipped with a topology, the so-called Krull topology, which in turn is central to generalize the above sketched connection of fields and groups to infinite field extensions. An advanced generalization of this idea, adapted to the needs of algebraic geometry, is the étale fundamental group.

Question: What concepts are fundamental to number theory? Question: What group uses infinite field extensions with topology? Question: What is used to generalize the connection of fields and groups to infinite field extensions? Question: What group is an advanced observation of infinite field extensions and groups that is adapted for the needs of algebraic geometry?
Lie groups are of fundamental importance in modern physics: Noether's theorem links continuous symmetries to conserved quantities. Rotation, as well as translations in space and time are basic symmetries of the laws of mechanics. They can, for instance, be used to construct simple models—imposing, say, axial symmetry on a situation will typically lead to significant simplification in the equations one needs to solve to provide a physical description.v[›] Another example are the Lorentz transformations, which relate measurements of time and velocity of two observers in motion relative to each other. They can be deduced in a purely group-theoretical way, by expressing the transformations as a rotational symmetry of Minkowski space. The latter serves—in the absence of significant gravitation—as a model of space time in special relativity. The full symmetry group of Minkowski space, i.e. including translations, is known as the Poincaré group. By the above, it plays a pivotal role in special relativity and, by implication, for quantum field theories. Symmetries that vary with location are central to the modern description of physical interactions with the help of gauge theory.

Question: What concept is of basic importance in modern physics? Question: What connects continuous symmetries to conserved quantities? Question: What term describes the basic symmetries of the laws of mechanics? Question: What concept relates measurements of time and velocity of two observers in motion relative to each other? Question: What describe the complete symmetry group of Minkowski space including translations?
In abstract algebra, more general structures are defined by relaxing some of the axioms defining a group. For example, if the requirement that every element has an inverse is eliminated, the resulting algebraic structure is called a monoid. The natural numbers N (including 0) under addition form a monoid, as do the nonzero integers under multiplication (Z ∖ {0}, ·), see above. There is a general method to formally add inverses to elements to any (abelian) monoid, much the same way as (Q ∖ {0}, ·) is derived from (Z ∖ {0}, ·), known as the Grothendieck group. Groupoids are similar to groups except that the composition a • b need not be defined for all a and b. They arise in the study of more complicated forms of symmetry, often in topological and analytical structures, such as the fundamental groupoid or stacks. Finally, it is possible to generalize any of these concepts by replacing the binary operation with an arbitrary n-ary one (i.e. an operation taking n arguments). With the proper generalization of the group axioms this gives rise to an n-ary group. The table gives a list of several structures generalizing groups.

Question: What product is created if the requirement that every element has an inverse is eliminated? Question: What type of numbers under addition form a monoid? Question: What method formally adds inverses to elements to any monoid? Question: What can be replaced to simplify abstract algebra concepts?
The Central African Republic (CAR; Sango: Ködörösêse tî Bêafrîka; French: République centrafricaine  pronounced: [ʁepyblik sɑ̃tʁafʁikɛn], or Centrafrique [sɑ̃tʀafʁik]) is a landlocked country in Central Africa. It is bordered by Chad to the north, Sudan to the northeast, South Sudan to the east, the Democratic Republic of the Congo and the Republic of the Congo to the south and Cameroon to the west. The CAR covers a land area of about 620,000 square kilometres (240,000 sq mi) and had an estimated population of around 4.7 million as of 2014[update].

Question: Where is the Central African Republic located? Question: What country borders CAR in the North? Question: What Country borders CAR in the South? Question: What Country borders CAR in the West? Question: What is the population of CAR approximately?
What is today the Central African Republic has been inhabited for millennia; however, the country's current borders were established by France, which ruled the country as a colony starting in the late 19th century. After gaining independence from France in 1960, the Central African Republic was ruled by a series of autocratic leaders; by the 1990s, calls for democracy led to the first multi-party democratic elections in 1993. Ange-Félix Patassé became president, but was later removed by General François Bozizé in the 2003 coup. The Central African Republic Bush War began in 2004 and, despite a peace treaty in 2007 and another in 2011, fighting broke out between various factions in December 2012, leading to ethnic and religious cleansing of the Muslim minority and massive population displacement in 2013 and 2014.

Question: Who established CAR's borders? Question: What country ruled CAR as a colony? Question: When did CAR become independent? Question: Who was the first democratically elected President? Question: What religion is a minority in CAR?
Approximately 10,000 years ago, desertification forced hunter-gatherer societies south into the Sahel regions of northern Central Africa, where some groups settled and began farming as part of the Neolithic Revolution. Initial farming of white yam progressed into millet and sorghum, and before 3000 BC the domestication of African oil palm improved the groups' nutrition and allowed for expansion of the local populations. Bananas arrived in the region and added an important source of carbohydrates to the diet; they were also used in the production of alcoholic beverages.[when?] This Agricultural Revolution, combined with a "Fish-stew Revolution", in which fishing began to take place, and the use of boats, allowed for the transportation of goods. Products were often moved in ceramic pots, which are the first known examples of artistic expression from the region's inhabitants.

Question: How long ago did people start living in the area? Question: What agricultural job did people settling in the area perform? Question: What vegetation helped populations expand? Question: What was an imported food that provided most of the carbohydrates? Question: What important protein helped with expansion of the inhabitants?
During the 16th and 17th centuries slave traders began to raid the region as part of the expansion of the Saharan and Nile River slave routes. Their captives were slaved and shipped to the Mediterranean coast, Europe, Arabia, the Western Hemisphere, or to the slave ports and factories along the West and North Africa or South the Ubanqui and Congo rivers. In the mid 19th century, the Bobangi people became major slave traders and sold their captives to the Americas using the Ubangi river to reach the coast. During the 18th century Bandia-Nzakara peoples established the Bangassou Kingdom along the Ubangi River.

Question: When did the slave trade hit the CAR region? Question: What is one area where captive slaves were sent? Question: Which people were major slave traders in the 19th century? Question: Who did the Bobangi sell slave to? Question: What river saw the establishment of a new Kingdom?
In 1920 French Equatorial Africa was established and Ubangi-Shari was administered from Brazzaville. During the 1920s and 1930s the French introduced a policy of mandatory cotton cultivation, a network of roads was built, attempts were made to combat sleeping sickness and Protestant missions were established to spread Christianity. New forms of forced labor were also introduced and a large number of Ubangians were sent to work on the Congo-Ocean Railway. Many of these forced laborers died of exhaustion, illness, or the poor conditions which claimed between 20% and 25% of the 127,000 workers.

Question: When was Equatorial Africa established? Question: Where was Equatorial African ran from? Question: What plant was mandated to be grown by the French? Question: What important infrastructure was built in the 1920s. Question: What religion was spread through CAR?
In September 1940, during the Second World War, pro-Gaullist French officers took control of Ubangi-Shari and General Leclerc established his headquarters for the Free French Forces in Bangui. In 1946 Barthélémy Boganda was elected with 9,000 votes to the French National Assembly, becoming the first representative for CAR in the French government. Boganda maintained a political stance against racism and the colonial regime but gradually became disheartened with the French political system and returned to CAR to establish the Movement for the Social Evolution of Black Africa (MESAN) in 1950.

Question: During what war did French officers setup headquarters in CAR? Question: Who was elected in 1946 as the first representative? Question: How many votes did Boganda get? Question: What was Bogandas platform that he ran for? Question: What caused Boganda to leave his elected post?
In the Ubangi-Shari Territorial Assembly election in 1957, MESAN captured 347,000 out of the total 356,000 votes, and won every legislative seat, which led to Boganda being elected president of the Grand Council of French Equatorial Africa and vice-president of the Ubangi-Shari Government Council. Within a year, he declared the establishment of the Central African Republic and served as the country's first prime minister. MESAN continued to exist, but its role was limited. After Boganda's death in a plane crash on 29 March 1959, his cousin, David Dacko, took control of MESAN and became the country's first president after the CAR had formally received independence from France. Dacko threw out his political rivals, including former Prime Minister and Mouvement d'évolution démocratique de l'Afrique centrale (MEDAC), leader Abel Goumba, whom he forced into exile in France. With all opposition parties suppressed by November 1962, Dacko declared MESAN as the official party of the state.

Question: What was the overwhelming amount of votes MESAN captured? Question: How many seats did MESAN win? Question: Who was elected president of the Grand Council? Question: How did Boganda die? Question: Who took over after Boganda died?
In April 1979, young students protested against Bokassa's decree that all school attendees would need to buy uniforms from a company owned by one of his wives. The government violently suppressed the protests, killing 100 children and teenagers. Bokassa himself may have been personally involved in some of the killings. In September 1979, France overthrew Bokassa and "restored" Dacko to power (subsequently restoring the name of the country to the Central African Republic). Dacko, in turn, was again overthrown in a coup by General André Kolingba on 1 September 1981.

Question: What caused a violent uprising? Question: How many were killed in the April 1979 protests? Question: Who overthrew Bokassa after this incident? Question: Who did France restore into power? Question: Who eventually overthrew Dacko?
By 1990, inspired by the fall of the Berlin Wall, a pro-democracy movement arose. Pressure from the United States, France, and from a group of locally represented countries and agencies called GIBAFOR (France, the USA, Germany, Japan, the EU, the World Bank, and the UN) finally led Kolingba to agree, in principle, to hold free elections in October 1992 with help from the UN Office of Electoral Affairs. After using the excuse of alleged irregularities to suspend the results of the elections as a pretext for holding on to power, President Kolingba came under intense pressure from GIBAFOR to establish a "Conseil National Politique Provisoire de la République" (Provisional National Political Council, CNPPR) and to set up a "Mixed Electoral Commission", which included representatives from all political parties.[citation needed]

Question: What was inspired by the fall of the Berlin wall? Question: What large country pressured CAR to hold elections? Question: When were free elections held? Question: Who helped with the Free Elections? Question: What excuse did Kolingba use to deny the election results?
When a second round of elections were finally held in 1993, again with the help of the international community coordinated by GIBAFOR, Ange-Félix Patassé won in the second round of voting with 53% of the vote while Goumba won 45.6%. Patassé's party, the Mouvement pour la Libération du Peuple Centrafricain (MLPC) or Movement for the Liberation of the Central African People, gained a simple but not an absolute majority of seats in parliament, which meant Patassé's party required coalition partners.[citation needed]

Question: When was a second democratic election held? Question: Who became the first democratically elected president? Question: What percentage did Patasse win with? Question: What was the winning party? Question: Did Patasse get a majority of the votes?
Patassé purged many of the Kolingba elements from the government and Kolingba supporters accused Patassé's government of conducting a "witch hunt" against the Yakoma. A new constitution was approved on 28 December 1994 but had little impact on the country's politics. In 1996–1997, reflecting steadily decreasing public confidence in the government's erratic behaviour, three mutinies against Patassé's administration were accompanied by widespread destruction of property and heightened ethnic tension. During this time (1996) the Peace Corps evacuated all its volunteers to neighboring Cameroon. To date, the Peace Corps has not returned to the Central African Republic. The Bangui Agreements, signed in January 1997, provided for the deployment of an inter-African military mission, to Central African Republic and re-entry of ex-mutineers into the government on 7 April 1997. The inter-African military mission was later replaced by a U.N. peacekeeping force (MINURCA).

Question: When was the new constitution for CAR approved? Question: When did a mutiny form against Patasse? Question: The mutinies caused what effects on the country? Question: Where did the peace corps evacuate to? Question: What agreement gave the authority for deployment of military intervention?
In the aftermath of the failed coup, militias loyal to Patassé sought revenge against rebels in many neighborhoods of Bangui and incited unrest including the murder of many political opponents. Eventually, Patassé came to suspect that General François Bozizé was involved in another coup attempt against him, which led Bozizé to flee with loyal troops to Chad. In March 2003, Bozizé launched a surprise attack against Patassé, who was out of the country. Libyan troops and some 1,000 soldiers of Bemba's Congolese rebel organization failed to stop the rebels and Bozizé's forces succeeded in overthrowing Patassé.[citation needed]

Question: The failed coup caused what effects? Question: Which General was involved in another coup? Question: Where did General Bozize flee to escape? Question: When did Bozize finally attempt his coup? Question: Who succeeded in finally overthrowing Patasse?
In 2004 the Central African Republic Bush War began as forces opposed to Bozizé took up arms against his government. In May 2005 Bozizé won a presidential election that excluded Patassé and in 2006 fighting continued between the government and the rebels. In November 2006, Bozizé's government requested French military support to help them repel rebels who had taken control of towns in the country's northern regions. Though the initially public details of the agreement pertained to logistics and intelligence, the French assistance eventually included strikes by Mirage jets against rebel positions.

Question: What war started in 2004? Question: In what way did Bozize made his rule legitimate? Question: When did CAR ask for assistance from the French against rebels? Question: What did the rebels take over? Question: How did the French assist against the rebels?
The Syrte Agreement in February and the Birao Peace Agreement in April 2007 called for a cessation of hostilities, the billeting of FDPC fighters and their integration with FACA, the liberation of political prisoners, integration of FDPC into government, an amnesty for the UFDR, its recognition as a political party, and the integration of its fighters into the national army. Several groups continued to fight but other groups signed on to the agreement, or similar agreements with the government (e.g. UFR on 15 December 2008). The only major group not to sign an agreement at the time was the CPJP, which continued its activities and signed a peace agreement with the government on 25 August 2012.

Question: What agreement called for the stop of Hostilities? Question: When was the Birao Peace Agreement signed? Question: What did the Birao Peace Agreement call for? Question: What major group did not sign the peace agreement immediately? Question: When did the CPJP finally sign the peace agreement?
Michel Djotodia took over as president and in May 2013 Central African Republic's Prime Minister Nicolas Tiangaye requested a UN peacekeeping force from the UN Security Council and on 31 May former President Bozizé was indicted for crimes against humanity and incitement of genocide. The security situation did not improve during June–August 2013 and there were reports of over 200,000 internally displaced persons (IDPs) as well as human rights abuses and renewed fighting between Séléka and Bozizé supporters.

Question: Who became president in 2013? Question: What was Bozize indicted for? Question: What mass murder did Bozize commit? Question: How many people were displaced in the unrests? Question: What faction was fighting against the remaining Bozize loyalists?