romikgosai/squad_modified huggingface dataset:
Sub- and quotient groups are related in the following way: a subset H of G can be seen as an injective map H → G, i.e. any element of the target has at most one element that maps to it. The counterpart to injective maps are surjective maps (every element of the target is mapped onto), such as the canonical map G → G / N.y[›] Interpreting subgroup and quotients in light of these homomorphisms emphasizes the structural concept inherent to these definitions alluded to in the introduction. In general, homomorphisms are neither injective nor surjective. Kernel and image of group homomorphisms and the first isomorphism theorem address this phenomenon.

Questions and answers:
['
Q: What map shows the relation between sub and quotient groups?
A: injective map', '
Q: What are the opposites of injective maps?
A: surjective maps', '
Q: What is an example of a surjective map?
A: canonical map', '
Q: What theory address the phenomenon of homomorphisms being neither injective nor surjective?
A: the first isomorphism theorem']
Groups are also applied in many other mathematical areas. Mathematical objects are often examined by associating groups to them and studying the properties of the corresponding groups. For example, Henri Poincaré founded what is now called algebraic topology by introducing the fundamental group. By means of this connection, topological properties such as proximity and continuity translate into properties of groups.i[›] For example, elements of the fundamental group are represented by loops. The second image at the right shows some loops in a plane minus a point. The blue loop is considered null-homotopic (and thus irrelevant), because it can be continuously shrunk to a point. The presence of the hole prevents the orange loop from being shrunk to a point. The fundamental group of the plane with a point deleted turns out to be infinite cyclic, generated by the orange loop (or any other loop winding once around the hole). This way, the fundamental group detects the hole.

Questions and answers:
['
Q: What are usually analyzed by associating groups to them and studying the elements of the corresponding groups?
A: Mathematical objects', '
Q: Who founded algebraic topology?
A: Henri Poincaré', '
Q: What did Henri Poincaré introduce when he established algebraic topology?
A: the fundamental group', '
Q: What topological properties translate into properties of groups?
A: proximity and continuity', '
Q: What represents elements of the fundamental group?
A: loops']
In modular arithmetic, two integers are added and then the sum is divided by a positive integer called the modulus. The result of modular addition is the remainder of that division. For any modulus, n, the set of integers from 0 to n − 1 forms a group under modular addition: the inverse of any element a is n − a, and 0 is the identity element. This is familiar from the addition of hours on the face of a clock: if the hour hand is on 9 and is advanced 4 hours, it ends up on 1, as shown at the right. This is expressed by saying that 9 + 4 equals 1 "modulo 12" or, in symbols,

Questions and answers:
['
Q: What positive integer is used to divide the sum of two positive integers in modular mathematics?
A: the modulus', '
Q: What results from modular addition?
A: the remainder of that division', '
Q: What type of device can be use to demonstrate modular addition?
A: a clock']
For any prime number p, there is also the multiplicative group of integers modulo p. Its elements are the integers 1 to p − 1. The group operation is multiplication modulo p. That is, the usual product is divided by p and the remainder of this division is the result of modular multiplication. For example, if p = 5, there are four group elements 1, 2, 3, 4. In this group, 4 · 4 = 1, because the usual product 16 is equivalent to 1, which divided by 5 yields a remainder of 1. for 5 divides 16 − 1 = 15, denoted

Questions and answers:
['
Q: What term describes the group of integers related to a prime number?
A: modulo p', '
Q: What integers are included in modulo p?
A: 1 to p − 1', '
Q: How many group elements exist if p=5?
A: four group elements']
In the groups Z/nZ introduced above, the element 1 is primitive, so these groups are cyclic. Indeed, each element is expressible as a sum all of whose terms are 1. Any cyclic group with n elements is isomorphic to this group. A second example for cyclic groups is the group of n-th complex roots of unity, given by complex numbers z satisfying zn = 1. These numbers can be visualized as the vertices on a regular n-gon, as shown in blue at the right for n = 6. The group operation is multiplication of complex numbers. In the picture, multiplying with z corresponds to a counter-clockwise rotation by 60°. Using some field theory, the group Fp× can be shown to be cyclic: for example, if p = 5, 3 is a generator since 31 = 3, 32 = 9 ≡ 4, 33 ≡ 2, and 34 ≡ 1.

Questions and answers:
['
Q: What is a group labeled when the element 1 is primitive?
A: cyclic', '
Q: What group is isomorphic to cyclic groups?
A: Any cyclic group with n elements', '
Q: What example of cyclic group satisfies the express of zn = 1?
A: the group of n-th complex roots of unity']
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.

Questions and answers:
['
Q: What are groups consisting of symmetries of given arithmetic concepts?
A: Symmetry groups', '
Q: What term describes the introductory symmetry group of the square?
A: geometric nature', '
Q: What can regarded as the study of symmetry?
A: group theory', '
Q: What is the name of the rule that must be met for a group operation to occur?
A: group law', '
Q: The group pattern is connected to the structure of the target by what behavior?
A: 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.

Questions and answers:
['
Q: What aids in predicting changes of physical traits?
A: group theory', '
Q: What stage of a physical transformation can group theory be used to make prediction?
A: phase transition', '
Q: What temperature causes the change of ferroelectric materials?
A: Curie temperature', '
Q: What term describes the vibrational lattice mode that turns to 0 frequency at the change?
A: 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.

Questions and answers:
['
Q: What are examples of finite symmetry groups used in coding theory?
A: Mathieu groups', '
Q: What is used for error correction of transferred data?
A: coding theory', '
Q: What describes functions having antiderivatives of a prescribed form?
A: differential Galois theory', '
Q: What concept investigates geometric elements that stay stable under group action?
A: 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.

Questions and answers:
['
Q: What groups combine matrices with matrix multiplication?
A: Matrix groups', '
Q: What includes all invertible n-by-n matrices with real entries?
A: The general linear group', '
Q: What term describes subgroups of the general linear group?
A: matrix groups', '
Q: What matrix group portrays all possible rotations in n dimensions?
A: the special orthogonal group SO(n)', '
Q: Rotation matrix groups are utilized in computer graphics with what concept?
A: 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.

Questions and answers:
['
Q: What degree does not include simple formulas for cubic and quatric equations?
A: degree 5 and higher', '
Q: What concept is associated with the solvability of polynomials?
A: Abstract properties of Galois groups', '
Q: What are used to express the solutions of polynomials?
A: 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.

Questions and answers:
['
Q: What does a finite group include?
A: a finite number of elements', '
Q: What is the number of elements in a group named?
A: the order of the group', '
Q: What type of class has a finite group that can be expressed as a subgroup of a symmetric group?
A: fundamental', '
Q: What can be described as the group of symmetries of an equilateral triangle?
A: 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.

Questions and answers:
['
Q: What level of finality do mathematicians try to reach with math concepts? 
A: complete classification', '
Q: What area of classification leads to troublesome arithmetic?
A: finite groups', "
Q: What defines finite groups of order p, a prime number, as being necessarily cyclic (abelian) groups Zp?
A: Lagrange's theorem", '
Q: What can be used to classify small groups even though there is no classification of all finite groups?
A: Computer algebra systems', '
Q: What describes finite simple groups as the building pieces for all finite groups?
A: 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:

Questions and answers:
['
Q: What group operations must occur for group law and topology to integrate well?
A: continuous functions', '
Q: What is the most simple example of topological groups?
A: reals R under addition, (R ∖ {0}, ·),', '
Q: What variables do locally compact groups share that can be studied by harmonic analysis?
A: 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.

Questions and answers:
['
Q: What concepts are fundamental to number theory?
A: adele rings and adelic algebraic groups', '
Q: What group uses infinite field extensions with topology?
A: the absolute Galois group', '
Q: What is used to generalize the connection of fields and groups to infinite field extensions?
A: Krull topology', '
Q: What group is an advanced observation of infinite field extensions and groups that is adapted for the needs of algebraic geometry?
A: 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.

Questions and answers:
['
Q: What concept is of basic importance in modern physics?
A: Lie groups', "
Q: What connects continuous symmetries to conserved quantities?
A: Noether's theorem", '
Q: What term describes the basic symmetries of the laws of mechanics?
A: Rotation', '
Q: What concept relates measurements of time and velocity of two observers in motion relative to each other?
A: Lorentz transformations', '
Q: What describe the complete symmetry group of Minkowski space including translations?
A: 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.

Questions and answers:
['
Q: What product is created if the requirement that every element has an inverse is eliminated?
A: monoid', '
Q: What type of numbers under addition form a monoid?
A: natural numbers N (including 0)', '
Q: What method formally adds inverses to elements to any monoid?
A: the Grothendieck group', '
Q: What can be replaced to simplify abstract algebra concepts?
A: 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].

Questions and answers:
['
Q: Where is the Central African Republic located?
A: Central Africa', '
Q: What country borders CAR in the North?
A: Chad', '
Q: What Country borders CAR in the South?
A: the Republic of the Congo', '
Q: What Country borders CAR in the West?
A: Cameroon', '
Q: What is the population of CAR approximately?
A: 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.

Questions and answers:
["
Q: Who established CAR's borders?
A: France", '
Q: What country ruled CAR as a colony?
A: France', '
Q: When did CAR become independent?
A: in 1960', '
Q: Who was the first democratically elected President?
A: Ange-Félix Patassé', '
Q: What religion is a minority in CAR?
A: 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.

Questions and answers:
['
Q: How long ago did people start living in the area?
A: Approximately 10,000 years ago', '
Q: What agricultural job did people settling in the area perform?
A: farming', '
Q: What vegetation helped populations expand?
A: African oil palm', '
Q: What was an imported food that provided most of the carbohydrates?
A: Bananas', '
Q: What important protein helped with expansion of the inhabitants?
A: 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.

Questions and answers:
['
Q: When did the slave trade hit the CAR region?
A: 16th and 17th centuries', '
Q: What is one area where captive slaves were sent?
A: Europe', '
Q: Which people were major slave traders in the 19th century?
A: Bobangi people', '
Q: Who did the Bobangi sell slave to?
A: the Americas', '
Q: What river saw the establishment of a new Kingdom?
A: 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.

Questions and answers:
['
Q: When was Equatorial Africa established?
A: 1920', '
Q: Where was Equatorial African ran from?
A: Brazzaville', '
Q: What plant was mandated to be grown by the French?
A: cotton', '
Q: What important infrastructure was built in the 1920s.
A: a network of roads', '
Q: What religion was spread through CAR?
A: Christianity']