romikgosai/squad_modified_2 huggingface dataset:
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.

Q1: What was King Juan Carlos' response to the media when asked if he would sign a bill supporting same-sex marriage?
A1: "Soy el Rey de España y no el de Bélgica"
Q2: What other controversial bill did the king give his assent to in 2010?
A2: a law permitting abortion on demand
Q3: When did same-sex marriages become legal in Spain?
A3: 3 July 2005
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.

Q1: If the Spanish monarch is deemed incapable of granting assent, who is the power transferred to?
A1: the next person of age in the line of succession
Q2: Could a Spanish ruler follow the same action of events as King Baudouin did in Belgium?
A2: not be possible under the current constitution
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.

Q1: Which articles allow the soverign to refise consent even when the bills have been passed through the Legislative assembly?
A1: 41 and 68
Q2: What statement by King George Tupou emphasized democracy?
A2: he would be guided by his prime minister in the exercising of his powers
Q3: In which year was the Arms and Ammunitions (Amemndment) Bill adopted?
A3: 2011
Q4: In what month did the King veto the Arms and Ammunitions Amendment Bill?
A4: December
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.

Q1: What is an arithmetical structure comprising of a set of elements including an operation that joins any two elements to form a third element?
A1: a group
Q2: What is the name used to describe the four conditions of closure, associativity, identity and invertibility?
A2: group axioms
Q3: What permits elements with different mathematical starting points to be taken care of in a flexible manner while holding their key basic aspects?
A3: abstract formalization of the group axioms
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.

Q1: What shares an essential relationship to groups?
A1: symmetry
Q2: What are the symmetry groups utilized as a part of the Standard Model particle physics?
A2: Lie groups
Q3: What groups are are utilized toward comprehending symmetry wonders in chemistry?
A3: Point groups
Q4: What groups can express the physical symmetry hidden behind special relativity?
A4: Poincaré groups
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.

Q1: Where did the idea of a group come from?
A1: the study of polynomial equations,
Q2: When was the group notion summed up and solidly settled?
A2: 1870
Q3: What are smaller and easier to understand groups broken down into?
A3: subgroups, quotient groups and simple groups.
Q4: When was announcement for the classification of finite simple groups?
A4: 1983
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.

Q1: What is known as underlying set of the group?
A1: The set
Q2: What is often utilized as a short name for the group?
A2: the group's underlying set
Q3: What expressions are utilized when is really implied to be a longer expression?
A3: shorthand expressions
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

Q1: What sends a point in the square to the relating point under the symmetry?
A1: functions
Q2:  What gives another symmetry function?
A2: Composing two of these symmetry functions
Q3: What group includes the symmetries of degree 4 and denoted D4?
A3: the dihedral group
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.

Q1:  Which French mathematician expanded on earlier work of Paolo Ruffini and Joseph-Louis Lagrange?
A1: Évariste Galois
Q2: The journey for answers to polynomial equations of degree higher than 4 was the original motivation for what theory?
A2: group theory
Q3:  Who developed a theory giving the first abstract definition of a finite group?
A3: Arthur Cayley
Q4: What modern concept was created from many fields of mathematics?
A4: abstract group
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.

Q1: When did the uniform theory of groups develop from different sources?
A1: 1870
Q2: Who presented a method for specifying a group by means of generators and relations?
A2: Walther von Dyck
Q3: What theory did Hermann Weyl study in addition to locally compact groups?
A3: The theory of Lie groups
Q4: Who initially molded the theory of algebraic groups?
A4: Claude Chevalley
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[›]

Q1: Where did the group of theorists first meet?
A1: The University of Chicago
Q2: What university class year belonged to the group of theorists?
A2: 1960–61
Q3: What did the group of theorists classify in 1982?
A3: all finite simple groups
Q4: What two factors describe the large size of the project?
A4: length of proof and number of researchers.
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.

Q1: What ideas are used to understand groups beyond symbols?
A1: structural concepts
Q2: What has to be compatible with the group operation?
A2: constructions related to groups
Q3: What concept describes groups that can be related to each other via functions?
A3: group homomorphisms
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.

Q1: What are two groups called if they include homomorphisms?
A1: isomorphic
Q2: What similar element do isomorphic groups carry?
A2: isomorphic groups
Q3: How can showing the second equality yields the first prove the concept of isomorphic groups?
A3: applying a to the first equality yields the second
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[›]

Q1: What is composed of two rotations?
A1: a rotation
Q2: What rotation can a rotation be reversed by?
A2: inverse
Q3: What essential condition must be met for a subset of a group to be a subgroup?
A3: The subgroup test
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

Q1: When is it preferable to consider two group elements to be the same?
A1: irrelevant to the question whether a reflection has been performed
Q2: What should not be considered when asking if a reflection has been performed?
A2: rotation operations
Q3: What number sets are used to show how subgroups can be seen as translations of the larger group?
A3: Cosets
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[›]

Q1: What is another term for coset multiplication?
A1: coset addition
Q2: What group operation is taken by the set from the original group?
A2: coset multiplication
Q3: What group does the inverse of gN include?
A3: quotient group
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

Q1: What groups can be combined to describe every group?
A1: is the quotient of the free group over the generators of the group
Q2: What group can be the quotient of the free group over the generators of the group?
A2: any group
Q3: What is a finite composition of two symmetries or their inverses?
A3: every symmetry of the square
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.

Q1: What map shows the relation between sub and quotient groups?
A1: injective map
Q2: What are the opposites of injective maps?
A2: surjective maps
Q3: What is an example of a surjective map?
A3: canonical map
Q4: What theory address the phenomenon of homomorphisms being neither injective nor surjective?
A4: 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.

Q1: What are usually analyzed by associating groups to them and studying the elements of the corresponding groups?
A1: Mathematical objects
Q2: Who founded algebraic topology?
A2: Henri Poincaré
Q3: What did Henri Poincaré introduce when he established algebraic topology?
A3: the fundamental group
Q4: What topological properties translate into properties of groups?
A4: proximity and continuity
Q5: What represents elements of the fundamental group?
A5: 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,

Q1: What positive integer is used to divide the sum of two positive integers in modular mathematics?
A1: the modulus
Q2: What results from modular addition?
A2: the remainder of that division
Q3: What type of device can be use to demonstrate modular addition?
A3: 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

Q1: What term describes the group of integers related to a prime number?
A1: modulo p
Q2: What integers are included in modulo p?
A2: 1 to p − 1
Q3: How many group elements exist if p=5?
A3: 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.

Q1: What is a group labeled when the element 1 is primitive?
A1: cyclic
Q2: What group is isomorphic to cyclic groups?
A2: Any cyclic group with n elements
Q3: What example of cyclic group satisfies the express of zn = 1?
A3: the group of n-th complex roots of unity