romikgosai/squad_modified_2 huggingface dataset:
Symmetry groups are groups consisting of symmetries of given mathematical objects—be they of geometric nature, such as the introductory symmetry group of the square, or of algebraic nature, such as polynomial equations and their solutions. Conceptually, group theory can be thought of as the study of symmetry.t[›] Symmetries in mathematics greatly simplify the study of geometrical or analytical objects. A group is said to act on another mathematical object X if every group element performs some operation on X compatibly to the group law. In the rightmost example below, an element of order 7 of the (2,3,7) triangle group acts on the tiling by permuting the highlighted warped triangles (and the other ones, too). By a group action, the group pattern is connected to the structure of the object being acted on.

Q1: What are groups consisting of symmetries of given arithmetic concepts?
A1: Symmetry groups
Q2: What term describes the introductory symmetry group of the square?
A2: geometric nature
Q3: What can regarded as the study of symmetry?
A3: group theory
Q4: What is the name of the rule that must be met for a group operation to occur?
A4: group law
Q5: The group pattern is connected to the structure of the target by what behavior?
A5: group action
Likewise, group theory helps predict the changes in physical properties that occur when a material undergoes a phase transition, for example, from a cubic to a tetrahedral crystalline form. An example is ferroelectric materials, where the change from a paraelectric to a ferroelectric state occurs at the Curie temperature and is related to a change from the high-symmetry paraelectric state to the lower symmetry ferroelectic state, accompanied by a so-called soft phonon mode, a vibrational lattice mode that goes to zero frequency at the transition.

Q1: What aids in predicting changes of physical traits?
A1: group theory
Q2: What stage of a physical transformation can group theory be used to make prediction?
A2: phase transition
Q3: What temperature causes the change of ferroelectric materials?
A3: Curie temperature
Q4: What term describes the vibrational lattice mode that turns to 0 frequency at the change?
A4: soft phonon mode
Finite symmetry groups such as the Mathieu groups are used in coding theory, which is in turn applied in error correction of transmitted data, and in CD players. Another application is differential Galois theory, which characterizes functions having antiderivatives of a prescribed form, giving group-theoretic criteria for when solutions of certain differential equations are well-behaved.u[›] Geometric properties that remain stable under group actions are investigated in (geometric) invariant theory.

Q1: What are examples of finite symmetry groups used in coding theory?
A1: Mathieu groups
Q2: What is used for error correction of transferred data?
A2: coding theory
Q3: What describes functions having antiderivatives of a prescribed form?
A3: differential Galois theory
Q4: What concept investigates geometric elements that stay stable under group action?
A4: invariant theory
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.

Q1: What groups combine matrices with matrix multiplication?
A1: Matrix groups
Q2: What includes all invertible n-by-n matrices with real entries?
A2: The general linear group
Q3: What term describes subgroups of the general linear group?
A3: matrix groups
Q4: What matrix group portrays all possible rotations in n dimensions?
A4: the special orthogonal group SO(n)
Q5: Rotation matrix groups are utilized in computer graphics with what concept?
A5: Euler angles
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.

Q1: What degree does not include simple formulas for cubic and quatric equations?
A1: degree 5 and higher
Q2: What concept is associated with the solvability of polynomials?
A2: Abstract properties of Galois groups
Q3: What are used to express the solutions of polynomials?
A3: radicals
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.

Q1: What does a finite group include?
A1: a finite number of elements
Q2: What is the number of elements in a group named?
A2: the order of the group
Q3: What type of class has a finite group that can be expressed as a subgroup of a symmetric group?
A3: fundamental
Q4: What can be described as the group of symmetries of an equilateral triangle?
A4: S3
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.

Q1: What level of finality do mathematicians try to reach with math concepts?
A1: complete classification
Q2: What area of classification leads to troublesome arithmetic?
A2: finite groups
Q3: What defines finite groups of order p, a prime number, as being necessarily cyclic (abelian) groups Zp?
A3: Lagrange's theorem
Q4: What can be used to classify small groups even though there is no classification of all finite groups?
A4: Computer algebra systems
Q5: What describes finite simple groups as the building pieces for all finite groups?
A5: The Jordan–Hölder theorem
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:

Q1: What group operations must occur for group law and topology to integrate well?
A1: continuous functions
Q2: What is the most simple example of topological groups?
A2: reals R under addition, (R ∖ {0}, ·),
Q3: What variables do locally compact groups share that can be studied by harmonic analysis?
A3: Haar measures
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.

Q1: What concepts are fundamental to number theory?
A1: adele rings and adelic algebraic groups
Q2: What group uses infinite field extensions with topology?
A2: the absolute Galois group
Q3: What is used to generalize the connection of fields and groups to infinite field extensions?
A3: Krull topology
Q4: What group is an advanced observation of infinite field extensions and groups that is adapted for the needs of algebraic geometry?
A4: the étale fundamental group
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.

Q1: What concept is of basic importance in modern physics?
A1: Lie groups
Q2: What connects continuous symmetries to conserved quantities?
A2: Noether's theorem
Q3: What term describes the basic symmetries of the laws of mechanics?
A3: Rotation
Q4: What concept relates measurements of time and velocity of two observers in motion relative to each other?
A4: Lorentz transformations
Q5: What describe the complete symmetry group of Minkowski space including translations?
A5: Poincaré group
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.

Q1: What product is created if the requirement that every element has an inverse is eliminated?
A1: monoid
Q2: What type of numbers under addition form a monoid?
A2: natural numbers N (including 0)
Q3: What method formally adds inverses to elements to any monoid?
A3: the Grothendieck group
Q4: What can be replaced to simplify abstract algebra concepts?
A4: the binary operation
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].

Q1: Where is the Central African Republic located?
A1: Central Africa
Q2: What country borders CAR in the North?
A2: Chad
Q3: What Country borders CAR in the South?
A3: the Republic of the Congo
Q4: What Country borders CAR in the West?
A4: Cameroon
Q5: What is the population of CAR approximately?
A5: 4.7 million
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.

Q1: Who established CAR's borders?
A1: France
Q2: What country ruled CAR as a colony?
A2: France
Q3: When did CAR become independent?
A3: in 1960
Q4: Who was the first democratically elected President?
A4: Ange-Félix Patassé
Q5: What religion is a minority in CAR?
A5: Muslim minority
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.

Q1: How long ago did people start living in the area?
A1: Approximately 10,000 years ago
Q2: What agricultural job did people settling in the area perform?
A2: farming
Q3: What vegetation helped populations expand?
A3: African oil palm
Q4: What was an imported food that provided most of the carbohydrates?
A4: Bananas
Q5: What important protein helped with expansion of the inhabitants?
A5: Fish
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.

Q1: When did the slave trade hit the CAR region?
A1: 16th and 17th centuries
Q2: What is one area where captive slaves were sent?
A2: Europe
Q3: Which people were major slave traders in the 19th century?
A3: Bobangi people
Q4: Who did the Bobangi sell slave to?
A4: the Americas
Q5: What river saw the establishment of a new Kingdom?
A5: Ubangi River
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.

Q1: When was Equatorial Africa established?
A1: 1920
Q2: Where was Equatorial African ran from?
A2: Brazzaville
Q3: What plant was mandated to be grown by the French?
A3: cotton
Q4: What important infrastructure was built in the 1920s.
A4: a network of roads
Q5: What religion was spread through CAR?
A5: Christianity
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.

Q1: During what war did French officers setup headquarters in CAR?
A1: Second World War
Q2: Who was elected in 1946 as the first representative?
A2: Barthélémy Boganda
Q3: How many votes did Boganda get?
A3: 9,000 votes
Q4: What was Bogandas platform that he ran for?
A4: political stance against racism
Q5: What caused Boganda to leave his elected post?
A5: disheartened with the French political system
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.

Q1: What was the overwhelming amount of votes MESAN captured?
A1: 347,000 out of the total 356,000 votes
Q2: How many seats did MESAN win?
A2: every legislative seat
Q3: Who was elected president of the Grand Council?
A3: Boganda
Q4: How did Boganda die?
A4: a plane crash
Q5: Who took over after Boganda died?
A5: his cousin, David Dacko
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.

Q1: What caused a violent uprising?
A1: young students protested against Bokassa's decree
Q2: How many were killed in the April 1979 protests?
A2: 100 children and teenagers
Q3: Who overthrew Bokassa after this incident?
A3: France
Q4: Who did France restore into power?
A4: Dacko
Q5: Who eventually overthrew Dacko?
A5: General André Kolingba
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]

Q1: What was inspired by the fall of the Berlin wall?
A1: a pro-democracy movement
Q2: What large country pressured CAR to hold elections?
A2: the United States
Q3: When were free elections held?
A3: October 1992
Q4: Who helped with the Free Elections?
A4: UN Office of Electoral Affairs
Q5: What excuse did Kolingba use to deny the election results?
A5: alleged irregularities
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]

Q1: When was a second democratic election held?
A1: in 1993
Q2: Who became the first democratically elected president?
A2: Ange-Félix Patassé
Q3: What percentage did Patasse win with?
A3: 53% of the vote
Q4: What was the winning party?
A4: Mouvement pour la Libération du Peuple Centrafricain
Q5: Did Patasse get a majority of the votes?
A5: gained a simple but not an absolute majority