BackpropBuff/adversarial_qa_adversarialQA huggingface dataset:
Question: In the equation, what was the lowercase g supposed to indicate?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[›]

Answer:elements
Question: What do you need information about to understand the entirety of a group?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[›]

Answer:subgroups
Question: In the equation, what was the notation used to indicate a subset?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[›]

Answer:H
Question: For what measurement is its complement the same number?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[›]

Answer:180°
Question: What is the inverse for 90?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[›]

Answer:270
Question: What information does the subgroup give?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[›]

Answer:the identity and the rotations
Question: What makes a subgroup necessary?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[›]

Answer:Knowing the subgroups is important in understanding the group as a whole
Question: Under what theory is the relationship between the structure of groups and the structure of fields being studied?for any constant c. Matrix groups over these fields fall under this regime, as do adele rings and adelic algebraic groups, which are basic to number theory. Galois groups of infinite field extensions such as the absolute Galois group can also be equipped with a topology, the so-called Krull topology, which in turn is central to generalize the above sketched connection of fields and groups to infinite field extensions. An advanced generalization of this idea, adapted to the needs of algebraic geometry, is the étale fundamental group.

Answer:Galois
Question: What do these values form?for any constant c. Matrix groups over these fields fall under this regime, as do adele rings and adelic algebraic groups, which are basic to number theory. Galois groups of infinite field extensions such as the absolute Galois group can also be equipped with a topology, the so-called Krull topology, which in turn is central to generalize the above sketched connection of fields and groups to infinite field extensions. An advanced generalization of this idea, adapted to the needs of algebraic geometry, is the étale fundamental group.

Answer:number theory
Question: How does the idea change?for any constant c. Matrix groups over these fields fall under this regime, as do adele rings and adelic algebraic groups, which are basic to number theory. Galois groups of infinite field extensions such as the absolute Galois group can also be equipped with a topology, the so-called Krull topology, which in turn is central to generalize the above sketched connection of fields and groups to infinite field extensions. An advanced generalization of this idea, adapted to the needs of algebraic geometry, is the étale fundamental group.

Answer:An advanced generalization of this idea, adapted to the needs of algebraic geometry, is the étale fundamental group
Question: What's part of the link?for any constant c. Matrix groups over these fields fall under this regime, as do adele rings and adelic algebraic groups, which are basic to number theory. Galois groups of infinite field extensions such as the absolute Galois group can also be equipped with a topology, the so-called Krull topology, which in turn is central to generalize the above sketched connection of fields and groups to infinite field extensions. An advanced generalization of this idea, adapted to the needs of algebraic geometry, is the étale fundamental group.

Answer:the so-called Krull topology
Question: What are these groups part of?for any constant c. Matrix groups over these fields fall under this regime, as do adele rings and adelic algebraic groups, which are basic to number theory. Galois groups of infinite field extensions such as the absolute Galois group can also be equipped with a topology, the so-called Krull topology, which in turn is central to generalize the above sketched connection of fields and groups to infinite field extensions. An advanced generalization of this idea, adapted to the needs of algebraic geometry, is the étale fundamental group.

Answer:algebraic geometry
Question: What is the theory's more improved form?for any constant c. Matrix groups over these fields fall under this regime, as do adele rings and adelic algebraic groups, which are basic to number theory. Galois groups of infinite field extensions such as the absolute Galois group can also be equipped with a topology, the so-called Krull topology, which in turn is central to generalize the above sketched connection of fields and groups to infinite field extensions. An advanced generalization of this idea, adapted to the needs of algebraic geometry, is the étale fundamental group.

Answer:the étale fundamental group
Question: What do these values make up?for any constant c. Matrix groups over these fields fall under this regime, as do adele rings and adelic algebraic groups, which are basic to number theory. Galois groups of infinite field extensions such as the absolute Galois group can also be equipped with a topology, the so-called Krull topology, which in turn is central to generalize the above sketched connection of fields and groups to infinite field extensions. An advanced generalization of this idea, adapted to the needs of algebraic geometry, is the étale fundamental group.

Answer:c. Matrix groups
Question: What is the theory applied to?for any constant c. Matrix groups over these fields fall under this regime, as do adele rings and adelic algebraic groups, which are basic to number theory. Galois groups of infinite field extensions such as the absolute Galois group can also be equipped with a topology, the so-called Krull topology, which in turn is central to generalize the above sketched connection of fields and groups to infinite field extensions. An advanced generalization of this idea, adapted to the needs of algebraic geometry, is the étale fundamental group.

Answer:the needs of algebraic geometry
Question: What do the Galois groups of certain types of field extensions associated with?for any constant c. Matrix groups over these fields fall under this regime, as do adele rings and adelic algebraic groups, which are basic to number theory. Galois groups of infinite field extensions such as the absolute Galois group can also be equipped with a topology, the so-called Krull topology, which in turn is central to generalize the above sketched connection of fields and groups to infinite field extensions. An advanced generalization of this idea, adapted to the needs of algebraic geometry, is the étale fundamental group.

Answer:field extensions
Question: What are matrix groups for any constant c?for any constant c. Matrix groups over these fields fall under this regime, as do adele rings and adelic algebraic groups, which are basic to number theory. Galois groups of infinite field extensions such as the absolute Galois group can also be equipped with a topology, the so-called Krull topology, which in turn is central to generalize the above sketched connection of fields and groups to infinite field extensions. An advanced generalization of this idea, adapted to the needs of algebraic geometry, is the étale fundamental group.

Answer:number theory
Question: What are these groups?for any constant c. Matrix groups over these fields fall under this regime, as do adele rings and adelic algebraic groups, which are basic to number theory. Galois groups of infinite field extensions such as the absolute Galois group can also be equipped with a topology, the so-called Krull topology, which in turn is central to generalize the above sketched connection of fields and groups to infinite field extensions. An advanced generalization of this idea, adapted to the needs of algebraic geometry, is the étale fundamental group.

Answer:Galois groups of infinite field extensions such as the absolute Galois group can also be equipped with a topology, the so-called Krull topology, which in turn is central to generalize the above sketched connection of fields and groups to infinite field extensions
Question: Where do these groups land?for any constant c. Matrix groups over these fields fall under this regime, as do adele rings and adelic algebraic groups, which are basic to number theory. Galois groups of infinite field extensions such as the absolute Galois group can also be equipped with a topology, the so-called Krull topology, which in turn is central to generalize the above sketched connection of fields and groups to infinite field extensions. An advanced generalization of this idea, adapted to the needs of algebraic geometry, is the étale fundamental group.

Answer:under this regime
Question: What is the Krull topology?for any constant c. Matrix groups over these fields fall under this regime, as do adele rings and adelic algebraic groups, which are basic to number theory. Galois groups of infinite field extensions such as the absolute Galois group can also be equipped with a topology, the so-called Krull topology, which in turn is central to generalize the above sketched connection of fields and groups to infinite field extensions. An advanced generalization of this idea, adapted to the needs of algebraic geometry, is the étale fundamental group.

Answer:Galois groups of infinite field extensions such as the absolute Galois group can also be equipped with a topology
Question: Who did Jordan work with?Mathematicians often strive for a complete classification (or list) of a mathematical notion. In the context of finite groups, this aim leads to difficult mathematics. According to Lagrange's theorem, finite groups of order p, a prime number, are necessarily cyclic (abelian) groups Zp. Groups of order p2 can also be shown to be abelian, a statement which does not generalize to order p3, as the non-abelian group D4 of order 8 = 23 above shows. Computer algebra systems can be used to list small groups, but there is no classification of all finite groups.q[›] An intermediate step is the classification of finite simple groups.r[›] A nontrivial group is called simple if its only normal subgroups are the trivial group and the group itself.s[›] The Jordan–Hölder theorem exhibits finite simple groups as the building blocks for all finite groups. Listing all finite simple groups was a major achievement in contemporary group theory. 1998 Fields Medal winner Richard Borcherds succeeded in proving the monstrous moonshine conjectures, a surprising and deep relation between the largest finite simple sporadic group—the "monster group"—and certain modular functions, a piece of classical complex analysis, and string theory, a theory supposed to unify the description of many physical phenomena.

Answer:Hölder
Question: What is Jordan occupation?Mathematicians often strive for a complete classification (or list) of a mathematical notion. In the context of finite groups, this aim leads to difficult mathematics. According to Lagrange's theorem, finite groups of order p, a prime number, are necessarily cyclic (abelian) groups Zp. Groups of order p2 can also be shown to be abelian, a statement which does not generalize to order p3, as the non-abelian group D4 of order 8 = 23 above shows. Computer algebra systems can be used to list small groups, but there is no classification of all finite groups.q[›] An intermediate step is the classification of finite simple groups.r[›] A nontrivial group is called simple if its only normal subgroups are the trivial group and the group itself.s[›] The Jordan–Hölder theorem exhibits finite simple groups as the building blocks for all finite groups. Listing all finite simple groups was a major achievement in contemporary group theory. 1998 Fields Medal winner Richard Borcherds succeeded in proving the monstrous moonshine conjectures, a surprising and deep relation between the largest finite simple sporadic group—the "monster group"—and certain modular functions, a piece of classical complex analysis, and string theory, a theory supposed to unify the description of many physical phenomena.

Answer:Mathematicians
Question: What is not p2?Mathematicians often strive for a complete classification (or list) of a mathematical notion. In the context of finite groups, this aim leads to difficult mathematics. According to Lagrange's theorem, finite groups of order p, a prime number, are necessarily cyclic (abelian) groups Zp. Groups of order p2 can also be shown to be abelian, a statement which does not generalize to order p3, as the non-abelian group D4 of order 8 = 23 above shows. Computer algebra systems can be used to list small groups, but there is no classification of all finite groups.q[›] An intermediate step is the classification of finite simple groups.r[›] A nontrivial group is called simple if its only normal subgroups are the trivial group and the group itself.s[›] The Jordan–Hölder theorem exhibits finite simple groups as the building blocks for all finite groups. Listing all finite simple groups was a major achievement in contemporary group theory. 1998 Fields Medal winner Richard Borcherds succeeded in proving the monstrous moonshine conjectures, a surprising and deep relation between the largest finite simple sporadic group—the "monster group"—and certain modular functions, a piece of classical complex analysis, and string theory, a theory supposed to unify the description of many physical phenomena.

Answer:a prime number
Question: What did Borcherds study?Mathematicians often strive for a complete classification (or list) of a mathematical notion. In the context of finite groups, this aim leads to difficult mathematics. According to Lagrange's theorem, finite groups of order p, a prime number, are necessarily cyclic (abelian) groups Zp. Groups of order p2 can also be shown to be abelian, a statement which does not generalize to order p3, as the non-abelian group D4 of order 8 = 23 above shows. Computer algebra systems can be used to list small groups, but there is no classification of all finite groups.q[›] An intermediate step is the classification of finite simple groups.r[›] A nontrivial group is called simple if its only normal subgroups are the trivial group and the group itself.s[›] The Jordan–Hölder theorem exhibits finite simple groups as the building blocks for all finite groups. Listing all finite simple groups was a major achievement in contemporary group theory. 1998 Fields Medal winner Richard Borcherds succeeded in proving the monstrous moonshine conjectures, a surprising and deep relation between the largest finite simple sporadic group—the "monster group"—and certain modular functions, a piece of classical complex analysis, and string theory, a theory supposed to unify the description of many physical phenomena.

Answer:contemporary group theory
Question: What is the main point of the passage?Mathematicians often strive for a complete classification (or list) of a mathematical notion. In the context of finite groups, this aim leads to difficult mathematics. According to Lagrange's theorem, finite groups of order p, a prime number, are necessarily cyclic (abelian) groups Zp. Groups of order p2 can also be shown to be abelian, a statement which does not generalize to order p3, as the non-abelian group D4 of order 8 = 23 above shows. Computer algebra systems can be used to list small groups, but there is no classification of all finite groups.q[›] An intermediate step is the classification of finite simple groups.r[›] A nontrivial group is called simple if its only normal subgroups are the trivial group and the group itself.s[›] The Jordan–Hölder theorem exhibits finite simple groups as the building blocks for all finite groups. Listing all finite simple groups was a major achievement in contemporary group theory. 1998 Fields Medal winner Richard Borcherds succeeded in proving the monstrous moonshine conjectures, a surprising and deep relation between the largest finite simple sporadic group—the "monster group"—and certain modular functions, a piece of classical complex analysis, and string theory, a theory supposed to unify the description of many physical phenomena.

Answer:Mathematicians often strive for a complete classification (or list) of a mathematical notion
Question: What is an important concept that is mentioned that is important to understand the passage?Mathematicians often strive for a complete classification (or list) of a mathematical notion. In the context of finite groups, this aim leads to difficult mathematics. According to Lagrange's theorem, finite groups of order p, a prime number, are necessarily cyclic (abelian) groups Zp. Groups of order p2 can also be shown to be abelian, a statement which does not generalize to order p3, as the non-abelian group D4 of order 8 = 23 above shows. Computer algebra systems can be used to list small groups, but there is no classification of all finite groups.q[›] An intermediate step is the classification of finite simple groups.r[›] A nontrivial group is called simple if its only normal subgroups are the trivial group and the group itself.s[›] The Jordan–Hölder theorem exhibits finite simple groups as the building blocks for all finite groups. Listing all finite simple groups was a major achievement in contemporary group theory. 1998 Fields Medal winner Richard Borcherds succeeded in proving the monstrous moonshine conjectures, a surprising and deep relation between the largest finite simple sporadic group—the "monster group"—and certain modular functions, a piece of classical complex analysis, and string theory, a theory supposed to unify the description of many physical phenomena.

Answer:Jordan–Hölder theorem
Question: What prevents simplification?Mathematicians often strive for a complete classification (or list) of a mathematical notion. In the context of finite groups, this aim leads to difficult mathematics. According to Lagrange's theorem, finite groups of order p, a prime number, are necessarily cyclic (abelian) groups Zp. Groups of order p2 can also be shown to be abelian, a statement which does not generalize to order p3, as the non-abelian group D4 of order 8 = 23 above shows. Computer algebra systems can be used to list small groups, but there is no classification of all finite groups.q[›] An intermediate step is the classification of finite simple groups.r[›] A nontrivial group is called simple if its only normal subgroups are the trivial group and the group itself.s[›] The Jordan–Hölder theorem exhibits finite simple groups as the building blocks for all finite groups. Listing all finite simple groups was a major achievement in contemporary group theory. 1998 Fields Medal winner Richard Borcherds succeeded in proving the monstrous moonshine conjectures, a surprising and deep relation between the largest finite simple sporadic group—the "monster group"—and certain modular functions, a piece of classical complex analysis, and string theory, a theory supposed to unify the description of many physical phenomena.

Answer:complete classification
Question: What did Lagrange study?Mathematicians often strive for a complete classification (or list) of a mathematical notion. In the context of finite groups, this aim leads to difficult mathematics. According to Lagrange's theorem, finite groups of order p, a prime number, are necessarily cyclic (abelian) groups Zp. Groups of order p2 can also be shown to be abelian, a statement which does not generalize to order p3, as the non-abelian group D4 of order 8 = 23 above shows. Computer algebra systems can be used to list small groups, but there is no classification of all finite groups.q[›] An intermediate step is the classification of finite simple groups.r[›] A nontrivial group is called simple if its only normal subgroups are the trivial group and the group itself.s[›] The Jordan–Hölder theorem exhibits finite simple groups as the building blocks for all finite groups. Listing all finite simple groups was a major achievement in contemporary group theory. 1998 Fields Medal winner Richard Borcherds succeeded in proving the monstrous moonshine conjectures, a surprising and deep relation between the largest finite simple sporadic group—the "monster group"—and certain modular functions, a piece of classical complex analysis, and string theory, a theory supposed to unify the description of many physical phenomena.

Answer:finite groups
Question: What is the main conclusion drawn by the passage?Mathematicians often strive for a complete classification (or list) of a mathematical notion. In the context of finite groups, this aim leads to difficult mathematics. According to Lagrange's theorem, finite groups of order p, a prime number, are necessarily cyclic (abelian) groups Zp. Groups of order p2 can also be shown to be abelian, a statement which does not generalize to order p3, as the non-abelian group D4 of order 8 = 23 above shows. Computer algebra systems can be used to list small groups, but there is no classification of all finite groups.q[›] An intermediate step is the classification of finite simple groups.r[›] A nontrivial group is called simple if its only normal subgroups are the trivial group and the group itself.s[›] The Jordan–Hölder theorem exhibits finite simple groups as the building blocks for all finite groups. Listing all finite simple groups was a major achievement in contemporary group theory. 1998 Fields Medal winner Richard Borcherds succeeded in proving the monstrous moonshine conjectures, a surprising and deep relation between the largest finite simple sporadic group—the "monster group"—and certain modular functions, a piece of classical complex analysis, and string theory, a theory supposed to unify the description of many physical phenomena.

Answer:the "monster group"—and certain modular functions, a piece of classical complex analysis, and string theory, a theory supposed to unify the description of many physical phenomena
Question: What is a part of the new state?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.

Answer:a so-called soft phonon mode
Question: What type of change happens at the Curie temperature when  the proper name for the change from a paraelectric to a ferroelectric state occursLikewise, 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.

Answer:changes in physical properties
Question: What is the proper name for the phase transition point at which 1060 degrees F is reachedLikewise, 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.

Answer:Curie temperature
Question: What is the proper name for the phase transition point at which  570 degrees C is reachedLikewise, 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.

Answer:Curie temperature
Question: Which of the following is not a potential state of ferroelectric materials: paraelectric, cubic or ferroelectric?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.

Answer:cubic
Question: What happen to the shape in the phase transition besides the change in crystalline form.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.

Answer:lower symmetry
Question: What happens to the material in the theory?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.

Answer:changes in physical properties that occur
Question: When does the change occur?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.

Answer:at the Curie temperature
Question: Which of the following is not a potential crystalline form: cubic, paraelectric or tetrahedral?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.

Answer:paraelectric
Question: Which physical property has a transition point named after a famous female scientist?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.

Answer:Curie
Question: Whats needed to understand a group by it's presentationQuotient 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

Answer:Quotient groups and subgroups
Question: What can describe a presentation of every groupQuotient 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

Answer:Quotient groups and subgroups
Question: What are quotient and subgroupsQuotient 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

Answer:a way of describing every group by its presentation
Question: what three words begin the passage?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

Answer:Quotient groups and
Question: what sentence end the passage?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

Answer:Together with the relations
Question: The example uses a right and vertical whatQuotient 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

Answer:reflection
Question: What is the free groupQuotient 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

Answer:subgroups
Question: what is been discussed?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

Answer:Quotient
Question: what was given as a sample?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

Answer:dihedral group D4
Question: If the square is symmetrical what does it meanQuotient 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

Answer:finite composition of these two symmetries or their inverses
Question: To whom was the right to vote extended in 1867?Palmerston died in 1865, and after a brief ministry led by Russell, Derby returned to power. In 1866, Victoria attended the State Opening of Parliament for the first time since Albert's death. The following year she supported the passing of the Reform Act 1867 which doubled the electorate by extending the franchise to many urban working men, though she was not in favour of votes for women. Derby resigned in 1868, to be replaced by Benjamin Disraeli, who charmed Victoria. "Everyone likes flattery," he said, "and when you come to royalty you should lay it on with a trowel." With the phrase "we authors, Ma'am", he complimented her. Disraeli's ministry only lasted a matter of months, and at the end of the year his Liberal rival, William Ewart Gladstone, was appointed prime minister. Victoria found Gladstone's demeanour far less appealing; he spoke to her, she is thought to have complained, as though she were "a public meeting rather than a woman".

Answer:many urban working men
Question: Between the last time that Victoria attended the State Opening of Parliament prior to 1866 and her attendance at the State Opening of Pariliament in 1866, who passed away?Palmerston died in 1865, and after a brief ministry led by Russell, Derby returned to power. In 1866, Victoria attended the State Opening of Parliament for the first time since Albert's death. The following year she supported the passing of the Reform Act 1867 which doubled the electorate by extending the franchise to many urban working men, though she was not in favour of votes for women. Derby resigned in 1868, to be replaced by Benjamin Disraeli, who charmed Victoria. "Everyone likes flattery," he said, "and when you come to royalty you should lay it on with a trowel." With the phrase "we authors, Ma'am", he complimented her. Disraeli's ministry only lasted a matter of months, and at the end of the year his Liberal rival, William Ewart Gladstone, was appointed prime minister. Victoria found Gladstone's demeanour far less appealing; he spoke to her, she is thought to have complained, as though she were "a public meeting rather than a woman".

Answer:Albert
Question: Who was treated like a public meeting, but not like a woman?Palmerston died in 1865, and after a brief ministry led by Russell, Derby returned to power. In 1866, Victoria attended the State Opening of Parliament for the first time since Albert's death. The following year she supported the passing of the Reform Act 1867 which doubled the electorate by extending the franchise to many urban working men, though she was not in favour of votes for women. Derby resigned in 1868, to be replaced by Benjamin Disraeli, who charmed Victoria. "Everyone likes flattery," he said, "and when you come to royalty you should lay it on with a trowel." With the phrase "we authors, Ma'am", he complimented her. Disraeli's ministry only lasted a matter of months, and at the end of the year his Liberal rival, William Ewart Gladstone, was appointed prime minister. Victoria found Gladstone's demeanour far less appealing; he spoke to her, she is thought to have complained, as though she were "a public meeting rather than a woman".

Answer:Victoria
Question: Russell followed?Palmerston died in 1865, and after a brief ministry led by Russell, Derby returned to power. In 1866, Victoria attended the State Opening of Parliament for the first time since Albert's death. The following year she supported the passing of the Reform Act 1867 which doubled the electorate by extending the franchise to many urban working men, though she was not in favour of votes for women. Derby resigned in 1868, to be replaced by Benjamin Disraeli, who charmed Victoria. "Everyone likes flattery," he said, "and when you come to royalty you should lay it on with a trowel." With the phrase "we authors, Ma'am", he complimented her. Disraeli's ministry only lasted a matter of months, and at the end of the year his Liberal rival, William Ewart Gladstone, was appointed prime minister. Victoria found Gladstone's demeanour far less appealing; he spoke to her, she is thought to have complained, as though she were "a public meeting rather than a woman".

Answer:Palmerston
Question: Who supported extending the franchise to many urban working men, but not to women?Palmerston died in 1865, and after a brief ministry led by Russell, Derby returned to power. In 1866, Victoria attended the State Opening of Parliament for the first time since Albert's death. The following year she supported the passing of the Reform Act 1867 which doubled the electorate by extending the franchise to many urban working men, though she was not in favour of votes for women. Derby resigned in 1868, to be replaced by Benjamin Disraeli, who charmed Victoria. "Everyone likes flattery," he said, "and when you come to royalty you should lay it on with a trowel." With the phrase "we authors, Ma'am", he complimented her. Disraeli's ministry only lasted a matter of months, and at the end of the year his Liberal rival, William Ewart Gladstone, was appointed prime minister. Victoria found Gladstone's demeanour far less appealing; he spoke to her, she is thought to have complained, as though she were "a public meeting rather than a woman".

Answer:Victoria
Question: Derby followed?Palmerston died in 1865, and after a brief ministry led by Russell, Derby returned to power. In 1866, Victoria attended the State Opening of Parliament for the first time since Albert's death. The following year she supported the passing of the Reform Act 1867 which doubled the electorate by extending the franchise to many urban working men, though she was not in favour of votes for women. Derby resigned in 1868, to be replaced by Benjamin Disraeli, who charmed Victoria. "Everyone likes flattery," he said, "and when you come to royalty you should lay it on with a trowel." With the phrase "we authors, Ma'am", he complimented her. Disraeli's ministry only lasted a matter of months, and at the end of the year his Liberal rival, William Ewart Gladstone, was appointed prime minister. Victoria found Gladstone's demeanour far less appealing; he spoke to her, she is thought to have complained, as though she were "a public meeting rather than a woman".

Answer:Russell
Question: What happened in 1868?Palmerston died in 1865, and after a brief ministry led by Russell, Derby returned to power. In 1866, Victoria attended the State Opening of Parliament for the first time since Albert's death. The following year she supported the passing of the Reform Act 1867 which doubled the electorate by extending the franchise to many urban working men, though she was not in favour of votes for women. Derby resigned in 1868, to be replaced by Benjamin Disraeli, who charmed Victoria. "Everyone likes flattery," he said, "and when you come to royalty you should lay it on with a trowel." With the phrase "we authors, Ma'am", he complimented her. Disraeli's ministry only lasted a matter of months, and at the end of the year his Liberal rival, William Ewart Gladstone, was appointed prime minister. Victoria found Gladstone's demeanour far less appealing; he spoke to her, she is thought to have complained, as though she were "a public meeting rather than a woman".

Answer:Derby resigned
Question: What happened after Albert's death?Palmerston died in 1865, and after a brief ministry led by Russell, Derby returned to power. In 1866, Victoria attended the State Opening of Parliament for the first time since Albert's death. The following year she supported the passing of the Reform Act 1867 which doubled the electorate by extending the franchise to many urban working men, though she was not in favour of votes for women. Derby resigned in 1868, to be replaced by Benjamin Disraeli, who charmed Victoria. "Everyone likes flattery," he said, "and when you come to royalty you should lay it on with a trowel." With the phrase "we authors, Ma'am", he complimented her. Disraeli's ministry only lasted a matter of months, and at the end of the year his Liberal rival, William Ewart Gladstone, was appointed prime minister. Victoria found Gladstone's demeanour far less appealing; he spoke to her, she is thought to have complained, as though she were "a public meeting rather than a woman".

Answer:a brief ministry led by Russell
Question: What year did the person who supported electorates doubling for working men go to Parliment?Palmerston died in 1865, and after a brief ministry led by Russell, Derby returned to power. In 1866, Victoria attended the State Opening of Parliament for the first time since Albert's death. The following year she supported the passing of the Reform Act 1867 which doubled the electorate by extending the franchise to many urban working men, though she was not in favour of votes for women. Derby resigned in 1868, to be replaced by Benjamin Disraeli, who charmed Victoria. "Everyone likes flattery," he said, "and when you come to royalty you should lay it on with a trowel." With the phrase "we authors, Ma'am", he complimented her. Disraeli's ministry only lasted a matter of months, and at the end of the year his Liberal rival, William Ewart Gladstone, was appointed prime minister. Victoria found Gladstone's demeanour far less appealing; he spoke to her, she is thought to have complained, as though she were "a public meeting rather than a woman".

Answer:1866