yz2022/T2P-DownstreamNLP huggingface dataset:
Group_(mathematics)

The modern concept of an abstract group developed out of several fields of mathematics. The original motivation for group theory was the quest for solutions of polynomial equations of degree higher than 4. The 19th-century French mathematician Évariste Galois, extending prior work of Paolo Ruffini and Joseph-Louis Lagrange, gave a criterion for the solvability of a particular polynomial equation in terms of the symmetry group of its roots (solutions). The elements of such a Galois group correspond to certain permutations of the roots. At first, Galois' ideas were rejected by his contemporaries, and published only posthumously. More general permutation groups were investigated in particular by Augustin Louis Cauchy. Arthur Cayley's On the theory of groups, as depending on the symbolic equation θn = 1 (1854) gives the first abstract definition of a finite group.

Question:  Who developed a theory giving the first abstract definition of a finite group?
Answer: Arthur Cayley
Group_(mathematics)

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,

Question: What positive integer is used to divide the sum of two positive integers in modular mathematics?
Answer: the modulus

Question: What results from modular addition?
Answer: the remainder of that division
Royal_assent

In Canada, the traditional ceremony for granting assent in parliament was regularly used until the 21st century, long after it had been discontinued in the United Kingdom and other Commonwealth realms. One result, conceived as part of a string of royal duties intended to demonstrate Canada's status as an independent kingdom, was that King George VI personally assented to nine bills of the Canadian parliament during the 1939 royal tour of Canada—85 years after his great-grandmother, Queen Victoria, had last granted royal assent personally in the United Kingdom. Under the Royal Assent Act 2002, however, the alternative practice of granting assent in writing, with each house being notified separately ( the Speaker of the Senate or a representative reads to the senators the letters from the governor general regarding the written declaration of Royal Assent), was brought into force. As the act also provides, royal assent is to be signified—by the governor general, or, more often, by a deputy, usually a Justice of the Supreme Court, at least twice each calendar year: for the first appropriation measure and for at least one other act, usually the first non-appropriation measure passed. However, the act provides that a grant of royal assent is not rendered invalid by a failure to employ the traditional ceremony where required.

Question: With what bill in Canada did granting assent by writing become effective?
Answer: Royal Assent Act 2002
Group_(mathematics)

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,

Question: What type of device can be use to demonstrate modular addition?
Answer: a clock
Group_(mathematics)

Quotient groups and subgroups together form a way of describing every group by its presentation: any group is the quotient of the free group over the generators of the group, quotiented by the subgroup of relations. The dihedral group D4, for example, can be generated by two elements r and f (for example, r = r1, the right rotation and f = fv the vertical (or any other) reflection), which means that every symmetry of the square is a finite composition of these two symmetries or their inverses. Together with the relations

Question: What group can be the quotient of the free group over the generators of the group?
Answer: any group
Royal_assent

In Canada, the traditional ceremony for granting assent in parliament was regularly used until the 21st century, long after it had been discontinued in the United Kingdom and other Commonwealth realms. One result, conceived as part of a string of royal duties intended to demonstrate Canada's status as an independent kingdom, was that King George VI personally assented to nine bills of the Canadian parliament during the 1939 royal tour of Canada—85 years after his great-grandmother, Queen Victoria, had last granted royal assent personally in the United Kingdom. Under the Royal Assent Act 2002, however, the alternative practice of granting assent in writing, with each house being notified separately ( the Speaker of the Senate or a representative reads to the senators the letters from the governor general regarding the written declaration of Royal Assent), was brought into force. As the act also provides, royal assent is to be signified—by the governor general, or, more often, by a deputy, usually a Justice of the Supreme Court, at least twice each calendar year: for the first appropriation measure and for at least one other act, usually the first non-appropriation measure passed. However, the act provides that a grant of royal assent is not rendered invalid by a failure to employ the traditional ceremony where required.

Question: Why was King George VI's personal assent in 1939 noteworthy?
Answer: 85 years after his great-grandmother, Queen Victoria, had last granted royal assent personally
Royal_assent

Articles 41 and 68 of the constitution empower the sovereign to withhold royal assent from bills adopted by the Legislative Assembly. In 2010, the kingdom moved towards greater democracy, with King George Tupou V saying that he would be guided by his prime minister in the exercising of his powers. Nonetheless, this does not preclude an independent royal decision to exercise a right of veto. In November 2011, the assembly adopted an Arms and Ammunitions (Amendment) Bill, which reduced the possible criminal sentences for the illicit possession of firearms. The bill was adopted by ten votes to eight. Two members of the assembly had recently been charged with the illicit possession of firearms. The Prime Minister, Lord Tuʻivakanō, voted in favour of the amendment. Members of the opposition denounced the bill and asked the King to veto it, which he did in December.

Question: In which year was the Arms and Ammunitions (Amemndment) Bill adopted?
Answer: 2011
Group_(mathematics)

Quotient groups and subgroups together form a way of describing every group by its presentation: any group is the quotient of the free group over the generators of the group, quotiented by the subgroup of relations. The dihedral group D4, for example, can be generated by two elements r and f (for example, r = r1, the right rotation and f = fv the vertical (or any other) reflection), which means that every symmetry of the square is a finite composition of these two symmetries or their inverses. Together with the relations

Question: What is a finite composition of two symmetries or their inverses?
Answer: every symmetry of the square

Question: What groups can be combined to describe every group?
Answer: is the quotient of the free group over the generators of the group
Group_(mathematics)

The modern concept of an abstract group developed out of several fields of mathematics. The original motivation for group theory was the quest for solutions of polynomial equations of degree higher than 4. The 19th-century French mathematician Évariste Galois, extending prior work of Paolo Ruffini and Joseph-Louis Lagrange, gave a criterion for the solvability of a particular polynomial equation in terms of the symmetry group of its roots (solutions). The elements of such a Galois group correspond to certain permutations of the roots. At first, Galois' ideas were rejected by his contemporaries, and published only posthumously. More general permutation groups were investigated in particular by Augustin Louis Cauchy. Arthur Cayley's On the theory of groups, as depending on the symbolic equation θn = 1 (1854) gives the first abstract definition of a finite group.

Question:  Which French mathematician expanded on earlier work of Paolo Ruffini and Joseph-Louis Lagrange?
Answer: Évariste Galois
Royal_assent

In Canada, the traditional ceremony for granting assent in parliament was regularly used until the 21st century, long after it had been discontinued in the United Kingdom and other Commonwealth realms. One result, conceived as part of a string of royal duties intended to demonstrate Canada's status as an independent kingdom, was that King George VI personally assented to nine bills of the Canadian parliament during the 1939 royal tour of Canada—85 years after his great-grandmother, Queen Victoria, had last granted royal assent personally in the United Kingdom. Under the Royal Assent Act 2002, however, the alternative practice of granting assent in writing, with each house being notified separately ( the Speaker of the Senate or a representative reads to the senators the letters from the governor general regarding the written declaration of Royal Assent), was brought into force. As the act also provides, royal assent is to be signified—by the governor general, or, more often, by a deputy, usually a Justice of the Supreme Court, at least twice each calendar year: for the first appropriation measure and for at least one other act, usually the first non-appropriation measure passed. However, the act provides that a grant of royal assent is not rendered invalid by a failure to employ the traditional ceremony where required.

Question: When did Canada finally cease to use the traditional ceremony for granting assent as regular practice?
Answer: the 21st century
Group_(mathematics)

The concept of a group arose from the study of polynomial equations, starting with Évariste Galois in the 1830s. After contributions from other fields such as number theory and geometry, the group notion was generalized and firmly established around 1870. Modern group theory—an active mathematical discipline—studies groups in their own right.a[›] To explore groups, mathematicians have devised various notions to break groups into smaller, better-understandable pieces, such as subgroups, quotient groups and simple groups. In addition to their abstract properties, group theorists also study the different ways in which a group can be expressed concretely (its group representations), both from a theoretical and a computational point of view. A theory has been developed for finite groups, which culminated with the classification of finite simple groups announced in 1983.aa[›] Since the mid-1980s, geometric group theory, which studies finitely generated groups as geometric objects, has become a particularly active area in group theory.

Question: When was the group notion summed up and solidly settled?
Answer: 1870
Royal_assent

In Belgium, the sanction royale has the same legal effect as royal assent; the Belgian constitution requires a theoretically possible refusal of royal sanction to be countersigned—as any other act of the monarch—by a minister responsible before the House of Representatives. The monarch promulgates the law, meaning that he or she formally orders that the law be officially published and executed. In 1990, when King Baudouin advised his cabinet he could not, in conscience, sign a bill decriminalising abortion (a refusal patently not covered by a responsible minister), the Council of Ministers, at the King's own request, declared Baudouin incapable of exercising his powers. In accordance with the Belgian constitution, upon the declaration of the sovereign's incapacity, the Council of Ministers assumed the powers of the head of state until parliament could rule on the King's incapacity and appoint a regent. The bill was then assented to by all members of the Council of Ministers "on behalf of the Belgian People". In a joint meeting, both houses of parliament declared the King capable of exercising his powers again the next day.

Question: Who declared the monarch incapable of acting upon an abortion decriminalization bill in 1990?
Answer: Council of Ministers

Question: Which King refused to sign an abortion decriminalization bill in 1990?
Answer: King Baudouin
Group_(mathematics)

The University of Chicago's 1960–61 Group Theory Year brought together group theorists such as Daniel Gorenstein, John G. Thompson and Walter Feit, laying the foundation of a collaboration that, with input from numerous other mathematicians, classified all finite simple groups in 1982. This project exceeded previous mathematical endeavours by its sheer size, in both length of proof and number of researchers. Research is ongoing to simplify the proof of this classification. These days, group theory is still a highly active mathematical branch, impacting many other fields.a[›]

Question: What university class year belonged to the group of theorists?
Answer: 1960–61

Question: Where did the group of theorists first meet?
Answer: The University of Chicago

Question: What did the group of theorists classify in 1982?
Answer: all finite simple groups

Question: What two factors describe the large size of the project?
Answer: length of proof and number of researchers.
Royal_assent

In Belgium, the sanction royale has the same legal effect as royal assent; the Belgian constitution requires a theoretically possible refusal of royal sanction to be countersigned—as any other act of the monarch—by a minister responsible before the House of Representatives. The monarch promulgates the law, meaning that he or she formally orders that the law be officially published and executed. In 1990, when King Baudouin advised his cabinet he could not, in conscience, sign a bill decriminalising abortion (a refusal patently not covered by a responsible minister), the Council of Ministers, at the King's own request, declared Baudouin incapable of exercising his powers. In accordance with the Belgian constitution, upon the declaration of the sovereign's incapacity, the Council of Ministers assumed the powers of the head of state until parliament could rule on the King's incapacity and appoint a regent. The bill was then assented to by all members of the Council of Ministers "on behalf of the Belgian People". In a joint meeting, both houses of parliament declared the King capable of exercising his powers again the next day.

Question: Who declared the King capable of excersizing power again after a brief lapse in 1990?
Answer: both houses of parliament
Royal_assent

Title IV of the 1978 Spanish constitution invests the Consentimiento Real (Royal Assent) and promulgation (publication) of laws with the monarch of Spain, while Title III, The Cortes Generales, Chapter 2, Drafting of Bills, outlines the method by which bills are passed. According to Article 91, within fifteen days of passage of a bill by the Cortes Generales, the sovereign shall give his or her assent and publish the new law. Article 92 invests the monarch with the right to call for a referendum, on the advice of the president of the government (commonly referred to in English as the prime minister) and the authorisation of the cortes.

Question: Which article in the Spanish constitution gives the monarch the right to ask for a referendum?
Answer: Article 92

Question: What chapter in the Spanish constitution describes how bills are to be passed?
Answer: Chapter 2, Drafting of Bills
Group_(mathematics)

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.

Question: What are examples of finite symmetry groups used in coding theory?
Answer: Mathieu groups

Question: What is used for error correction of transferred data?
Answer: coding theory
Group_(mathematics)

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?
Answer: Matrix groups

Question: What includes all invertible n-by-n matrices with real entries?
Answer: The general linear group
Group_(mathematics)

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.

Question: What concept investigates geometric elements that stay stable under group action?
Answer: invariant theory

Question: What describes functions having antiderivatives of a prescribed form?
Answer: differential Galois theory
Group_(mathematics)

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 matrix group portrays all possible rotations in n dimensions?
Answer: the special orthogonal group SO(n)

Question: Rotation matrix groups are utilized in computer graphics with what concept?
Answer: Euler angles

Question: What term describes subgroups of the general linear group?
Answer: matrix groups
Group_(mathematics)

The convergence of these various sources into a uniform theory of groups started with Camille Jordan's Traité des substitutions et des équations algébriques (1870). Walther von Dyck (1882) introduced the idea of specifying a group by means of generators and relations, and was also the first to give an axiomatic definition of an "abstract group", in the terminology of the time. As of the 20th century, groups gained wide recognition by the pioneering work of Ferdinand Georg Frobenius and William Burnside, who worked on representation theory of finite groups, Richard Brauer's modular representation theory and Issai Schur's papers. The theory of Lie groups, and more generally locally compact groups was studied by Hermann Weyl, Élie Cartan and many others. Its algebraic counterpart, the theory of algebraic groups, was first shaped by Claude Chevalley (from the late 1930s) and later by the work of Armand Borel and Jacques Tits.

Question: When did the uniform theory of groups develop from different sources? 
Answer: 1870
Royal_assent

Articles 77–79 of the Norwegian Constitution specifically grant the monarch of Norway the right to withhold royal assent from any bill passed by the Storting. Should the sovereign ever choose to exercise this privilege, Article 79 provides a means by which his veto may be over-ridden: "If a Bill has been passed unaltered by two sessions of the Storting, constituted after two separate successive elections and separated from each other by at least two intervening sessions of the Storting, without a divergent Bill having been passed by any Storting in the period between the first and last adoption, and it is then submitted to the King with a petition that His Majesty shall not refuse his assent to a Bill which, after the most mature deliberation, the Storting considers to be beneficial, it shall become law even if the Royal Assent is not accorded before the Storting goes into recess."

Question: Which articles in the Norwegian Constitution allow the monarch to reject a bill that has been passed through the Sorting?
Answer: Articles 77–79
Group_(mathematics)

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.

Question: What aids in predicting changes of physical traits?
Answer: group theory
Group_(mathematics)

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.

Question: What group is isomorphic to cyclic groups?
Answer: Any cyclic group with n elements
Group_(mathematics)

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.

Question: What stage of a physical transformation can group theory be used to make prediction?
Answer: phase transition