Divyanshu67/squad_v2 huggingface dataset:
Royal_assent

The Royal Assent ceremony takes place in the Senate, as the sovereign is traditionally barred from the House of Commons. On the day of the event, the Speaker of the Senate will read to the chamber a notice from the secretary to the governor general indicating when the viceroy or a deputy thereof will arrive. The Senate thereafter cannot adjourn until after the ceremony. The speaker moves to sit beside the throne, the Mace Bearer, with mace in hand, stands adjacent to him or her, and the governor general enters to take the speaker's chair. The Usher of the Black Rod is then commanded by the speaker to summon the Members of Parliament, who follow Black Rod back to the Senate, the Sergeant-at-Arms carrying the mace of the House of Commons. In the Senate, those from the commons stand behind the bar, while Black Rod proceeds to stand next to the governor general, who then nods his or her head to signify Royal Assent to the presented bills (which do not include appropriations bills). Once the list of bills is complete, the Clerk of the Senate states: "in Her Majesty's name, His [or Her] Excellency the Governor General [or the deputy] doth assent to these bills." If there are any appropriation bills to receive Royal Assent, the Speaker of the House of Commons will read their titles and the Senate clerk repeats them to the governor general, who nods his or her head to communicate Royal Assent. When these bills have all been assented to, the Clerk of the Senate recites "in Her Majesty's name, His [or Her] Excellency the Governor General [or the deputy] thanks her loyal subjects, accepts their benevolence and assents to these bills. The governor general or his or her deputy then depart parliament.

Question: From where is the sovereign typically forbidden?
Answer: House of Commons

Question: Whose job is it to release parliament after the ceremony?
Answer: The governor general or his or her deputy

Question: Who reads the formal statement after the list of bills has been finished?
Answer: Clerk of the Senate

Question: Which position nods their head to signify assention?
Answer: governor general

Question: What ceremony takes place in the House?

Question: On the day of the event, the President of the Senate reads to whom?

Question: On the day of the event, the President of the Senate reads what?

Question: Who follows Black Rod back to the Senate?
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: What is royal assent called in Belgium?
Answer: sanction royale

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

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

Question: In Bulgaria, what has the same effect as royal assent?

Question: King Bedoin refused to sign a bill in what year?

Question: Both houses declared King Baudouin incapable of exercising his powers in what type of meeting?

Question: King Baudouin signed a bill criminalizing abortion in what year?
Royal_assent

The constitution of Jordan grants its monarch the right to withhold assent to laws passed by its parliament. Article 93 of that document gives the Jordanian sovereign six months to sign or veto any legislation sent to him from the National Assembly; if he vetoes it within that timeframe, the assembly may override his veto by a two-thirds vote of both houses; otherwise, the law does not go into effect (but it may be reconsidered in the next session of the assembly). If the monarch fails to act within six months of the bill being presented to him, it becomes law without his signature.

Question: How much time does a Jordinian leader have to sign or veto legislation?
Answer: six months

Question: What majority is required to override a veto from the soverign?
Answer: two-thirds

Question: What happens if the soverign doesn't sign the bill within the six-month time frame?
Answer: , it becomes law without his signature

Question: The constitution of Jordan does not grant the monarch the right to do what?

Question: Article 83 gives the Jordanian sovereign how much time to signor veto legislation?

Question: What kind of vote from either house is required to override the Jordanian sovereign?

Question: If the monarch fails to act within four months what happens?
Royal_assent

After the House of Representatives has debated the law, it either approves it and sends it to the Senate with the text "The Second Chamber of the States General sends the following approved proposal of law to the First Chamber", or it rejects it and returns it to the government with the text "The Second Chamber of the States General has rejected the accompanying proposal of law." If the upper house then approves the law, it sends it back to the government with the text "To the King, The States General have accepted the proposal of law as it is offered here."

Question: What text is written and sent to the Senate if the House approves of a law?
Answer: "The Second Chamber of the States General sends the following approved proposal of law to the First Chamber"

Question: What text is written and sent back to the government if a bill is rejected?
Answer: The Second Chamber of the States General has rejected the accompanying proposal of law."

Question: If the Senate approves the law after receiving it from the House, what is written and sent to the government?
Answer: "To the King, The States General have accepted the proposal of law as it is offered here."

Question: What is one thing that could happen after the House of Lords debates the law?

Question: If the lower house approves the law, what happens?

Question: What text is sent to the House of Representatives if a law is approved?

Question: What text is sent to the House of Representatives if a law is not approved?
Royal_assent

The government, consisting of the monarch and the ministers, will then usually approve the proposal and the sovereign and one of the ministers signs the proposal with the addition of an enacting clause, thereafter notifying the States General that "The King assents to the proposal." It has happened in exceptional circumstances that the government does not approve a law that has been passed in parliament. In such a case, neither the monarch nor a minister will sign the bill, notifying the States General that "The King will keep the proposal under advisement." A law that has received royal assent will be published in the State Magazine, with the original being kept in the archives of the King's Offices.

Question: Where are laws published after receiving assent?
Answer: the State Magazine

Question: Where is the original law text stored?
Answer: in the archives of the King's Offices

Question: What happens in the rare case the government doesn't approve of a law that has passed both houses?
Answer: neither the monarch nor a minister will sign the bill, notifying the States General that "The King will keep the proposal under advisement."

Question: The sovereign and two what signs the proposal?

Question: The enabling clause is added to what?

Question: In what case will the monarch but not the minister sign the bill?

Question: A law that has not received royal assent will be published where?
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

Question: Which article in the Norweigan Constitution provides a way to override a monarch's veto?
Answer: Article 79

Question: How many elections are required to assend a bill that the monarch has vetoed?
Answer: two

Question: Articles 67-69 of the Norwegian Constitution grant what?

Question: Which article provides a means by which a sovereigns' veto may not be over-ridden?

Question: Which articles of the Norwegian Constitution do not grant the monarch the right to withhold royal assent?

Question: The monarch of New England can withhold royal assent according to which articles?
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: Which article specifies the number of days available to the monarch to provide a signature on an assended bill?
Answer: Article 91

Question: In the 1978 Spanish constitution, which title describes how bills are passed?
Answer: Title III, The Cortes Generales

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

Question: Which title outlines the method by which bills are not passed?

Question: Which title does not invest the royal assent and publication of laws with the monarch of Spain?

Question: Title II, which describes how bills are passed was created in what year?

Question: The sovereign has 25 days to give his or her assent according to what article?

Question: The monarch does not have a right to call for a referendum according to which article?
Royal_assent

No provision within the constitution grants the monarch an ability to veto legislation directly; however, no provision prohibits the sovereign from withholding royal assent, which effectively constitutes a veto. When the Spanish media asked King Juan Carlos if he would endorse the bill legalising same-sex marriages, he answered "Soy el Rey de España y no el de Bélgica" ("I am the King of Spain and not that of Belgium")—a reference to King Baudouin I of Belgium, who had refused to sign the Belgian law legalising abortion. The King gave royal assent to Law 13/2005 on 1 July 2005; the law was gazetted in the Boletín Oficial del Estado on 2 July and came into effect on 3 July 2005. Likewise, in 2010, King Juan Carlos gave royal assent to a law permitting abortion on demand.

Question: What was King Juan Carlos' response to the media when asked if he would sign a bill supporting same-sex marriage?
Answer: "Soy el Rey de España y no el de Bélgica"

Question: What other controversial bill did the king give his assent to in 2010?
Answer: a law permitting abortion on demand

Question: When did same-sex marriages become legal in Spain?
Answer: 3 July 2005
Royal_assent

If the Spanish monarch ever refused in conscience to grant royal assent, a procedure similar to the Belgian handling of King Baudouin's objection would not be possible under the current constitution. If the sovereign were ever declared incapable of discharging royal authority, his or her powers would not be transferred to the Cabinet, pending the parliamentary appointment of a regency. Instead, the constitution mandates the next person of age in the line of succession would immediately become regent. Therefore, had Juan Carlos followed the Belgian example in 2005 or 2010, a declaration of incapacity would have transferred power to Felipe, then the heir apparent.

Question: If the Spanish monarch is deemed incapable of granting assent, who is the power transferred to?
Answer: the next person of age in the line of succession

Question: Could a Spanish ruler follow the same action of events as King Baudouin did in Belgium?
Answer: not be possible under the current constitution

Question: If the Spanish sovereign was found capable of discharging royal authority, what happens?

Question: If Juan Carlos had been declared capable, who would have been given power?

Question: Who is the current heir apparent to Juan Carlos?

Question: Who gets power if the sovereign were declared capable?
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: Which articles allow the soverign to refise consent even when the bills have been passed through the Legislative assembly?
Answer: 41 and 68

Question: What statement by King George Tupou emphasized democracy?
Answer: he would be guided by his prime minister in the exercising of his powers

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

Question: In what month did the King veto the Arms and Ammunitions Amendment Bill?
Answer: December

Question: What articles empower the sovereign to grant royal assent?

Question: The kingdom moved away from democracy in what year?

Question: An Art and Ammunitions Bill was adopted in what year?

Question: What bill was adopted by eight votes to ten?

Question: How many member of the assembly were charge with legal possession of firearms?
Group_(mathematics)

In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element. The operation satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility. One of the most familiar examples of a group is the set of integers together with the addition operation, but the abstract formalization of the group axioms, detached as it is from the concrete nature of any particular group and its operation, applies much more widely. It allows entities with highly diverse mathematical origins in abstract algebra and beyond to be handled in a flexible way while retaining their essential structural aspects. The ubiquity of groups in numerous areas within and outside mathematics makes them a central organizing principle of contemporary mathematics.

Question: What is an arithmetical structure comprising of a set of elements including an operation that joins any two elements to form a third element?
Answer: a group

Question: What is the name used to describe the four conditions of closure, associativity, identity and invertibility?
Answer: group axioms

Question: What permits elements with different mathematical starting points to be taken care of in a flexible manner while holding their key basic aspects?
Answer: abstract formalization of the group axioms

Question: What is a set of elements constructed from?

Question: What are the two elements called?

Question: What is the abstract formalization attached to?

Question: What do entities lose when handled in abstract algebra?

Question: What is an example of a rare group?
Group_(mathematics)

Groups share a fundamental kinship with the notion of symmetry. For example, a symmetry group encodes symmetry features of a geometrical object: the group consists of the set of transformations that leave the object unchanged and the operation of combining two such transformations by performing one after the other. Lie groups are the symmetry groups used in the Standard Model of particle physics; Point groups are used to help understand symmetry phenomena in molecular chemistry; and Poincaré groups can express the physical symmetry underlying special relativity.

Question: What shares an essential relationship to groups?
Answer: symmetry

Question: What are the symmetry groups utilized as a part of the Standard Model particle physics?
Answer: Lie groups

Question: What groups are are utilized toward comprehending symmetry wonders in chemistry? 
Answer: Point groups

Question: What groups can express the physical symmetry hidden behind special relativity?
Answer: Poincaré groups

Question: What encodes a symmetry group?

Question: What are groups dissimilar to?

Question: Point groups are used in what form of physics?

Question: Poincare groups are used to understand molecular what?

Question: What do Lie groups express in terms of special relativity?
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: Where did the idea of a group come from?
Answer: the study of polynomial equations,

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

Question: What are smaller and easier to understand groups broken down into?
Answer: subgroups, quotient groups and simple groups.

Question: When was announcement for the classification of finite simple groups?
Answer: 1983

Question: Who generalized group notion in 1870?

Question: What year did mathematicians begin studying groups in their own right?

Question: What are the two subgroups?

Question: When did theorists begin exploring the theoretical and computational point of view?

Question: When did the geometric group theory become less active?
Group_(mathematics)

The set G is called the underlying set of the group (G, •). Often the group's underlying set G is used as a short name for the group (G, •). Along the same lines, shorthand expressions such as "a subset of the group G" or "an element of group G" are used when what is actually meant is "a subset of the underlying set G of the group (G, •)" or "an element of the underlying set G of the group (G, •)". Usually, it is clear from the context whether a symbol like G refers to a group or to an underlying set.

Question: What is known as underlying set of the group?
Answer: The set

Question: What is often utilized as a short name for the group?
Answer: the group's underlying set

Question: What expressions are utilized when is really implied to be a longer expression?
Answer: shorthand expressions

Question: What cannot be used as the short name for the group?

Question: What is the overlaying set of the group?

Question: What can be unclear for the G symbol?

Question: A subset of the underlying set G of the group (G) cannot be written using what kind of expression?
Group_(mathematics)

These symmetries are represented by functions. Each of these functions sends a point in the square to the corresponding point under the symmetry. For example, r1 sends a point to its rotation 90° clockwise around the square's center, and fh sends a point to its reflection across the square's vertical middle line. Composing two of these symmetry functions gives another symmetry function. These symmetries determine a group called the dihedral group of degree 4 and denoted D4. The underlying set of the group is the above set of symmetry functions, and the group operation is function composition. Two symmetries are combined by composing them as functions, that is, applying the first one to the square, and the second one to the result of the first application. The result of performing first a and then b is written symbolically from right to left as

Question: What sends a point in the square to the relating point under the symmetry? 
Answer: functions

Question:  What gives another symmetry function?
Answer: Composing two of these symmetry functions

Question: What group includes the symmetries of degree 4 and denoted D4?
Answer: the dihedral group

Question: What are represented by symmetries?

Question: r1 is part of what group?

Question: Both symmetries are applied to what?

Question: What is written from left to right?

Question: How many degrees is fh rotated?
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: What modern concept was created from many fields of mathematics?
Answer: abstract group

Question: The journey for answers to polynomial equations of degree higher than 4 was the original motivation for what theory?
Answer: group theory

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

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

Question: People were looking for polynomial equations under what number?

Question: Whose work did Paolo Ruffini and Joseph-Louis Lagrange base their work on?

Question: What was Augustin Louis Cauchy's nationality?

Question: When did Arthur Cayley publish On the theory of groups?

Question: What equation did Galois create that created an abstract definition?
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

Question: Who presented a method for specifying a group by means of generators and relations?
Answer: Walther von Dyck

Question: What theory did Hermann Weyl study in addition to locally compact groups?
Answer: The theory of Lie groups

Question: Who initially molded the theory of algebraic groups?
Answer: Claude Chevalley

Question: What did Walter von Dyck publish in 1870?

Question: Who wrote the theory of Lie groups?

Question: What did Jacques Tits shape first?

Question: What was Issai Schur's theory on?

Question: Who worked on representation theory of lie groups?
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: Where did the group of theorists first meet?
Answer: The University of Chicago

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

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.

Question: What was classified in 1960?

Question: Who founded the University of Chicago's Group Theory Year?

Question: What no longer has much impact on other fields?

Question: What was shorter about this project as compared to other endeavors?

Question: When did the University of Chicago begin?
Group_(mathematics)

To understand groups beyond the level of mere symbolic manipulations as above, more structural concepts have to be employed.c[›] There is a conceptual principle underlying all of the following notions: to take advantage of the structure offered by groups (which sets, being "structureless", do not have), constructions related to groups have to be compatible with the group operation. This compatibility manifests itself in the following notions in various ways. For example, groups can be related to each other via functions called group homomorphisms. By the mentioned principle, they are required to respect the group structures in a precise sense. The structure of groups can also be understood by breaking them into pieces called subgroups and quotient groups. The principle of "preserving structures"—a recurring topic in mathematics throughout—is an instance of working in a category, in this case the category of groups.

Question: What ideas are used to understand groups beyond symbols?
Answer: structural concepts

Question: What has to be compatible with the group operation?
Answer: constructions related to groups

Question: What concept describes groups that can be related to each other via functions?
Answer: group homomorphisms

Question: What is needed to understand structural concepts?

Question: Both groups and sets have what?

Question: What function is used to break groups into pieces?

Question: Groups do not need to respect structures in what sense?

Question: What is an example of a subgroup?
Group_(mathematics)

Two groups G and H are called isomorphic if there exist group homomorphisms a: G → H and b: H → G, such that applying the two functions one after another in each of the two possible orders gives the identity functions of G and H. That is, a(b(h)) = h and b(a(g)) = g for any g in G and h in H. From an abstract point of view, isomorphic groups carry the same information. For example, proving that g • g = 1G for some element g of G is equivalent to proving that a(g) ∗ a(g) = 1H, because applying a to the first equality yields the second, and applying b to the second gives back the first.

Question: What are two groups called if they include homomorphisms?
Answer: isomorphic

Question: What similar element do isomorphic groups carry?
Answer: isomorphic groups

Question: How can showing the second equality yields the first prove the concept of isomorphic groups?
Answer: applying a to the first equality yields the second

Question: What are two groups called if no group homomorphisms are found?

Question: Isomorphic groups carry different what?

Question: What gives the identity of a?

Question: From what point of view do isomorphic groups have different information?

Question: How does the second equality disprove the concept of isomorphic groups?
Group_(mathematics)

In the example above, the identity and the rotations constitute a subgroup R = {id, r1, r2, r3}, highlighted in red in the group table above: any two rotations composed are still a rotation, and a rotation can be undone by (i.e. is inverse to) the complementary rotations 270° for 90°, 180° for 180°, and 90° for 270° (note that rotation in the opposite direction is not defined). The subgroup test is a necessary and sufficient condition for a subset H of a group G to be a subgroup: it is sufficient to check that g−1h ∈ H for all elements g, h ∈ H. Knowing the subgroups is important in understanding the group as a whole.d[›]

Question: What is composed of two rotations?
Answer: a rotation

Question: What rotation can a rotation be reversed by?
Answer: inverse

Question: What essential condition must be met for a subset of a group to be a subgroup?
Answer: The subgroup test

Question: What is defined when moving in the opposite direction?

Question: 270 for 180 is an example of what kind of rotation?

Question: subgroup R is made up of the inverse and what?

Question: What is unimportant for understanding the group as a whole?

Question: Knowing the group as a whole is important for understanding what?
Group_(mathematics)

In many situations it is desirable to consider two group elements the same if they differ by an element of a given subgroup. For example, in D4 above, once a reflection is performed, the square never gets back to the r2 configuration by just applying the rotation operations (and no further reflections), i.e. the rotation operations are irrelevant to the question whether a reflection has been performed. Cosets are used to formalize this insight: a subgroup H defines left and right cosets, which can be thought of as translations of H by arbitrary group elements g. In symbolic terms, the left and right cosets of H containing g are

Question: When is it preferable to consider two group elements to be the same?
Answer: irrelevant to the question whether a reflection has been performed

Question: What should not be considered when asking if a reflection has been performed?
Answer: rotation operations

Question: What number sets are used to show how subgroups can be seen as translations of the larger group?
Answer: Cosets

Question: When are two group elements considered different?

Question: In D4 the square can return to the r2 configuration by applying only what?

Question: What question are cosets irrelevant to?

Question: What does group element g define?

Question: When are rotation operations considered?
Group_(mathematics)

This set inherits a group operation (sometimes called coset multiplication, or coset addition) from the original group G: (gN) • (hN) = (gh)N for all g and h in G. This definition is motivated by the idea (itself an instance of general structural considerations outlined above) that the map G → G / N that associates to any element g its coset gN be a group homomorphism, or by general abstract considerations called universal properties. The coset eN = N serves as the identity in this group, and the inverse of gN in the quotient group is (gN)−1 = (g−1)N.e[›]

Question: What is another term for coset multiplication?
Answer: coset addition

Question: What group operation is taken by the set from the original group?
Answer: coset multiplication

Question: What group does the inverse of gN include?
Answer: quotient group

Question: What is the original group sometimes called?

Question: What are group homomorphisms called?

Question: What does the Coset gN serve as?

Question: What group operation is taken by the original group from the set?

Question: The inverse of gN excludes what group?
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 groups can be combined to describe every group?
Answer: is the quotient of the free group over the generators of the group

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

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

Question: What do quotient groups describe by themselves?

Question: What presentation describes every group?

Question: How many elements does the dihedral group generate?

Question: What of a square is an infinite composition?

Question: Which two elements does D4 generate?
Group_(mathematics)

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.

Question: What map shows the relation between sub and quotient groups?
Answer: injective map

Question: What are the opposites of injective maps?
Answer: surjective maps

Question: What is an example of a surjective map?
Answer: canonical map

Question: What theory address the phenomenon of homomorphisms being neither injective nor surjective?
Answer: the first isomorphism theorem

Question: Subset G of H is seen as what map?

Question: What is the minimum number of elements in a target?

Question: What kind of map is similar to an injective map?

Question: What theory discusses the surjective nature of canonical maps?

Question: What two forms must homomorphisms take?
Group_(mathematics)

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.

Question: What are usually analyzed by associating groups to them and studying the elements of the corresponding groups?
Answer: Mathematical objects

Question: Who founded algebraic topology?
Answer: Henri Poincaré

Question: What did Henri Poincaré introduce when he established algebraic topology?
Answer: the fundamental group

Question: What topological properties translate into properties of groups?
Answer: proximity and continuity

Question: What represents elements of the fundamental group?
Answer: loops

Question: Who first applied groups to other mathematical areas?

Question: Which topological properties do not carry over into groups?

Question: Why is the blue loop considered relevant?

Question: What causes the orange loop to shrink to a point?

Question: What results in a finite cycle?
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

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

Question: How many integers are divided in modular arithmetic?

Question: What are the two integers called?

Question: What is the result of the remained of the division?

Question: What is 9 in modular addition?

Question: If the hour hand is on 1 and is advanced by 4 hours, where does it end up?
Group_(mathematics)

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

Question: What term describes the group of integers related to a prime number?
Answer: modulo p

Question: What integers are included in modulo p?
Answer: 1 to p − 1

Question: How many group elements exist if p=5?
Answer: four group elements

Question: What describes a group of integers related to a modulo?

Question: What is the result of modular multiplication divided by?

Question: How many integers does p = 5 have?

Question: What is p divided by?

Question: What is 16 multiplied by 5 If p=5?
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 is a group labeled when the element 1 is primitive?
Answer: cyclic

Question: What group is isomorphic to cyclic groups?
Answer: Any cyclic group with n elements

Question: What example of cyclic group satisfies the express of zn = 1?
Answer: the group of n-th complex roots of unity

Question: A cyclic group without n elements is considered what?

Question: What element is not primitive in groups Z/nZ?

Question: What are groups called when element 1 is not primitive?

Question: What rotation results in multiplying but n?

Question: What are numbers on an irregular n-gon?
Group_(mathematics)

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.

Question: What are groups consisting of symmetries of given arithmetic concepts?
Answer: Symmetry groups

Question: What term describes the introductory symmetry group of the square?
Answer: geometric nature

Question: What can regarded as the study of symmetry?
Answer: group theory

Question: What is the name of the rule that must be met for a group operation to occur?
Answer: group law

Question: The group pattern is connected to the structure of the target by what behavior?
Answer: group action

Question: What do symmetry groups make?

Question: Which equation is are used for the square?

Question: What is the study of geometrical or analytical objects called?

Question: What makes the study of symmetries easier?

Question: What does group action do to the tiling?
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

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

Question: What temperature causes the change of ferroelectric materials?
Answer: Curie temperature

Question: What term describes the vibrational lattice mode that turns to 0 frequency at the change?
Answer: soft phonon mode

Question: What predicts the changes in group theory?

Question: Which stage can physical properties be used to make predictions about group theory?

Question: What temperature causes a change from ferroelectric to paraelectric?

Question: What is the frequency of a paraelectric state before transition?

Question: What is the name of the mode that moves material from a low-symmetry state to a high one?
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

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

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

Question: What theory is error correction applied to?

Question: Which form of finite symmetry groups are useless in coding theory?

Question: Mathieu groups have what kind of prescribed form?

Question: What stable properties does the Galois theory study?

Question: Geometric Invariant theory studies what kind of equations?
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

Question: What term describes subgroups of the general linear group?
Answer: matrix groups

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 are matrices and matrix groups combined called?

Question: The general linear group describes all possible what?

Question: What is an example of a large matrix group?

Question: What are subgroups of the special orthogonal group called?

Question: What type of angles are made of all invertible n-by-b matrices?